Define a continuous function. Continuity of functions - theorems and properties

Continuity of function. Breaking points.

The bull walks, sways, sighs as he goes:
- Oh, the board is running out, now I’m going to fall!

In this lesson we will examine the concept of continuity of a function, the classification of discontinuity points and a common practical problem continuity studies of functions. From the very name of the topic, many intuitively guess what will be discussed and think that the material is quite simple. This is true. But it is simple tasks that are most often punished for neglect and a superficial approach to solving them. Therefore, I recommend that you study the article very carefully and catch all the subtleties and techniques.

What do you need to know and be able to do? Not very much. To learn the lesson well, you need to understand what it is limit of a function. For readers with a low level of preparation, it is enough to comprehend the article Function limits. Examples of solutions and look at the geometric meaning of the limit in the manual Graphs and properties of elementary functions. It is also advisable to familiarize yourself with geometric transformations of graphs, since practice in most cases involves constructing a drawing. The prospects are optimistic for everyone, and even a full kettle will be able to cope with the task on its own in the next hour or two!

Continuity of function. Breakpoints and their classification

Concept of continuity of function

Let's consider some function that is continuous on the entire number line:

Or, to put it more succinctly, our function is continuous on (the set of real numbers).

What is the “philistine” criterion of continuity? Obviously, the graph of a continuous function can be drawn without lifting the pencil from the paper.

In this case, two simple concepts should be clearly distinguished: domain of a function And continuity of function. In general it's not the same thing. For example:

This function is defined on the entire number line, that is, for everyone The meaning of “x” has its own meaning of “y”. In particular, if , then . Note that the other point is punctuated, because by the definition of a function, the value of the argument must correspond to the only thing function value. Thus, domain our function: .

However this function is not continuous on ! It is quite obvious that at the point she is suffering gap. The term is also quite intelligible and visual; indeed, here the pencil will have to be torn off the paper anyway. A little later we will look at the classification of breakpoints.

Continuity of a function at a point and on an interval

In a particular mathematical problem, we can talk about the continuity of a function at a point, the continuity of a function on an interval, a half-interval, or the continuity of a function on a segment. That is, there is no “mere continuity”– the function can be continuous SOMEWHERE. And the fundamental “building block” of everything else is continuity of function at the point .

The theory of mathematical analysis gives a definition of the continuity of a function at a point using “delta” and “epsilon” neighborhoods, but in practice there is a different definition in use, to which we will pay close attention.

First let's remember one-sided limits who burst into our lives in the first lesson about function graphs. Consider an everyday situation:

If we approach the axis to the point left(red arrow), then the corresponding values of the “games” will go along the axis to the point (crimson arrow). Mathematically, this fact is fixed using left-hand limit:

Pay attention to the entry (reads “x tends to ka on the left”). The “additive” “minus zero” symbolizes , essentially this means that we are approaching the number from the left side.

Similarly, if you approach the point “ka” on right(blue arrow), then the “games” will come to the same value, but along the green arrow, and right-hand limit will be formatted as follows:

"Additive" symbolizes , and the entry reads: “x tends to ka on the right.”

If one-sided limits are finite and equal(as in our case): , then we will say that there is a GENERAL limit. It's simple, the general limit is our “usual” limit of a function, equal to a finite number.

Note that if the function is not defined at (poke out the black dot on the graph branch), then the above calculations remain valid. As has already been noted several times, in particular in the article on infinitesimal functions, expressions mean that "x" infinitely close approaches the point, while DOESN'T MATTER, whether the function itself is defined at a given point or not. A good example will be found in the next paragraph, when the function is analyzed.

Definition: a function is continuous at a point if the limit of the function at a given point is equal to the value of the function at that point: .

The definition is detailed in the following terms:

1) The function must be defined at the point, that is, the value must exist.

2) There must be a general limit of the function. As noted above, this implies the existence and equality of one-sided limits: .

3) The limit of the function at a given point must be equal to the value of the function at this point: .

If violated at least one of the three conditions, then the function loses the property of continuity at the point .

Continuity of a function over an interval is formulated ingeniously and very simply: a function is continuous on the interval if it is continuous at every point of the given interval.

In particular, many functions are continuous on an infinite interval, that is, on the set of real numbers. This is a linear function, polynomials, exponential, sine, cosine, etc. And in general, any elementary function continuous on its domain of definition, for example, a logarithmic function is continuous on the interval . Hopefully by now you have a pretty good idea of what graphs of basic functions look like. More detailed information about their continuity can be obtained from a kind man named Fichtenholtz.

With the continuity of a function on a segment and half-intervals, everything is also not difficult, but it is more appropriate to talk about this in class about finding the minimum and maximum values of a function on a segment, but for now let’s not worry about it.

Classification of break points

The fascinating life of functions is rich in all sorts of special points, and break points are only one of the pages of their biography.

Note : just in case, I’ll dwell on an elementary point: the breaking point is always single point– there are no “several break points in a row”, that is, there is no such thing as a “break interval”.

These points, in turn, are divided into two large groups: ruptures of the first kind And ruptures of the second kind. Each type of gap has its own characteristic features, which we will look at right now:

Discontinuity point of the first kind

If the continuity condition is violated at a point and one-sided limits finite , then it is called discontinuity point of the first kind.

Let's start with the most optimistic case. According to the original idea of the lesson, I wanted to tell the theory “in general terms,” but in order to demonstrate the reality of the material, I settled on the option with specific characters.

It’s sad, like a photo of newlyweds against the backdrop of the Eternal Flame, but the following shot is generally accepted. Let us depict the graph of the function in the drawing:

This function is continuous on the entire number line, except for the point. And in fact, the denominator cannot be equal to zero. However, in accordance with the meaning of the limit, we can infinitely close approach “zero” both from the left and from the right, that is, one-sided limits exist and, obviously, coincide:
(Condition No. 2 of continuity is satisfied).

But the function is not defined at the point, therefore, Condition No. 1 of continuity is violated, and the function suffers a discontinuity at this point.

A break of this type (with the existing general limit) are called repairable gap. Why removable? Because the function can redefine at the breaking point:

Does it look strange? Maybe. But such a function notation does not contradict anything! Now the gap is closed and everyone is happy:

Let's perform a formal check:

2) – there is a general limit;
3)

Thus, all three conditions are satisfied, and the function is continuous at a point by the definition of continuity of a function at a point.

However, matan haters can define the function in a bad way, for example :

It is interesting that the first two continuity conditions are satisfied here:
1) – the function is defined at a given point;
2) – there is a general limit.

But the third boundary has not been passed: , that is, the limit of the function at the point not equal the value of a given function at a given point.

Thus, at a point the function suffers a discontinuity.

The second, sadder case is called rupture of the first kind with a jump. And sadness is evoked by one-sided limits that finite and different. An example is shown in the second drawing of the lesson. Such a gap usually occurs when piecewise defined functions, which have already been mentioned in the article about graph transformations.

Consider the piecewise function and we will complete its drawing. How to build a graph? Very simple. On a half-interval we draw a fragment of a parabola (green), on an interval - a straight line segment (red) and on a half-interval - a straight line (blue).

Moreover, due to the inequality, the value is determined for the quadratic function (green dot), and due to the inequality, the value is determined for the linear function (blue dot):

In the most difficult case, you should resort to point-by-point construction of each piece of the graph (see the first lesson about graphs of functions).

Now we will only be interested in the point. Let's examine it for continuity:

2) Let's calculate one-sided limits.

On the left we have a red line segment, so the left-sided limit is:

On the right is the blue straight line, and the right-hand limit:

As a result, we received finite numbers, and they not equal. Since one-sided limits finite and different: , then our function tolerates discontinuity of the first kind with a jump.

It is logical that the gap cannot be eliminated - the function really cannot be further defined and “glued together”, as in the previous example.

Discontinuity points of the second kind

Usually, all other cases of rupture are cleverly classified into this category. I won’t list everything, because in practice, in 99% of problems you will encounter endless gap– when left-handed or right-handed, and more often, both limits are infinite.

And, of course, the most obvious picture is the hyperbola at point zero. Here both one-sided limits are infinite: , therefore, the function suffers a discontinuity of the second kind at the point .

I try to fill my articles with as diverse content as possible, so let's look at the graph of a function that has not yet been encountered:

according to the standard scheme:

1) The function is not defined at this point because the denominator goes to zero.

Of course, we can immediately conclude that the function suffers a discontinuity at point , but it would be good to classify the nature of the discontinuity, which is often required by the condition. For this:

Let me remind you that by recording we mean infinitesimal negative number, and under the entry - infinitesimal positive number.

One-sided limits are infinite, which means that the function suffers a discontinuity of the 2nd kind at the point . The y-axis is vertical asymptote for the graph.

It is not uncommon for both one-sided limits to exist, but only one of them is infinite, for example:

This is the graph of the function.

We examine the point for continuity:

1) The function is not defined at this point.

2) Let's calculate one-sided limits:

We will talk about the method of calculating such one-sided limits in the last two examples of the lecture, although many readers have already seen and guessed everything.

The left-hand limit is finite and equal to zero (we “do not go to the point itself”), but the right-hand limit is infinite and the orange branch of the graph approaches infinitely close to its vertical asymptote, given by the equation (black dotted line).

So the function suffers second kind discontinuity at point .

As for a discontinuity of the 1st kind, the function can be defined at the discontinuity point itself. For example, for a piecewise function Feel free to put a black bold dot at the origin of coordinates. On the right is a branch of a hyperbola, and the right-hand limit is infinite. I think almost everyone has an idea of what this graph looks like.

What everyone was looking forward to:

How to examine a function for continuity?

The study of a function for continuity at a point is carried out according to an already established routine scheme, which consists of checking three conditions of continuity:

Example 1

Explore function

Solution:

1) The only point within the scope is where the function is not defined.

2) Let's calculate one-sided limits:

One-sided limits are finite and equal.

Thus, at the point the function suffers a removable discontinuity.

What does the graph of this function look like?

I would like to simplify , and it seems like an ordinary parabola is obtained. BUT the original function is not defined at point , so the following clause is required:

Let's make the drawing:

Answer: the function is continuous on the entire number line except the point at which it suffers a removable discontinuity.

The function can be further defined in a good or not so good way, but according to the condition this is not required.

You say this is a far-fetched example? Not at all. This has happened dozens of times in practice. Almost all of the site’s tasks come from real independent work and tests.

Let's get rid of our favorite modules:

Example 2

Explore function for continuity. Determine the nature of the function discontinuities, if they exist. Execute the drawing.

Solution: For some reason, students are afraid and don’t like functions with a module, although there is nothing complicated about them. We have already touched on such things a little in the lesson. Geometric transformations of graphs. Since the module is non-negative, it is expanded as follows: , where “alpha” is some expression. In this case, and our function should be written piecewise:

But the fractions of both pieces must be reduced by . The reduction, as in the previous example, will not take place without consequences. The original function is not defined at the point since the denominator goes to zero. Therefore, the system should additionally specify the condition , and make the first inequality strict:

Now about a VERY USEFUL decision technique: before finalizing the task on a draft, it is advantageous to make a drawing (regardless of whether it is required by the conditions or not). This will help, firstly, to immediately see points of continuity and points of discontinuity, and, secondly, it will 100% protect you from errors when finding one-sided limits.

Let's do the drawing. In accordance with our calculations, to the left of the point it is necessary to draw a fragment of a parabola (blue color), and to the right - a piece of a parabola (red color), while the function is not defined at the point itself:

If in doubt, take a few x values and plug them into the function (remembering that the module destroys the possible minus sign) and check the graph.

Let us examine the function for continuity analytically:

1) The function is not defined at the point, so we can immediately say that it is not continuous at it.

2) Let’s establish the nature of the discontinuity; to do this, we calculate one-sided limits:

The one-sided limits are finite and different, which means that the function suffers a discontinuity of the 1st kind with a jump at the point . Note again that when finding limits, it does not matter whether the function at the break point is defined or not.

Now all that remains is to transfer the drawing from the draft (it was made as if with the help of research ;-)) and complete the task:

Answer: the function is continuous on the entire number line except for the point at which it suffers a discontinuity of the first kind with a jump.

Sometimes they require additional indication of the discontinuity jump. It is calculated simply - from the right limit you need to subtract the left limit: , that is, at the break point our function jumped 2 units down (as the minus sign tells us).

Example 3

Explore function for continuity. Determine the nature of the function discontinuities, if they exist. Make a drawing.

This is an example for you to solve on your own, a sample solution at the end of the lesson.

Let's move on to the most popular and widespread version of the task, when the function consists of three parts:

Example 4

Examine a function for continuity and plot a graph of the function .

Solution: it is obvious that all three parts of the function are continuous on the corresponding intervals, so it remains to check only two points of “junction” between the pieces. First, let's make a draft drawing; I commented on the construction technique in sufficient detail in the first part of the article. The only thing is that we need to carefully follow our singular points: due to the inequality, the value belongs to the straight line (green dot), and due to the inequality, the value belongs to the parabola (red dot):

Well, in principle, everything is clear =) All that remains is to formalize the decision. For each of the two “joining” points, we standardly check 3 continuity conditions:

I) We examine the point for continuity

The one-sided limits are finite and different, which means that the function suffers a discontinuity of the 1st kind with a jump at the point .

Let us calculate the discontinuity jump as the difference between the right and left limits:
, that is, the graph jerked up one unit.

II) We examine the point for continuity

1) – the function is defined at a given point.

2) Find one-sided limits:

– one-sided limits are finite and equal, which means there is a general limit.

3) – the limit of a function at a point is equal to the value of this function at a given point.

At the final stage, we transfer the drawing to the final version, after which we put the final chord:

Answer: the function is continuous on the entire number line, except for the point at which it suffers a discontinuity of the first kind with a jump.

Example 5

Examine a function for continuity and construct its graph .

This is an example for independent solution, a short solution and an approximate sample of the problem at the end of the lesson.

You may get the impression that at one point the function must be continuous, and at another there must be a discontinuity. In practice, this is not always the case. Try not to neglect the remaining examples - there will be several interesting and important features:

Example 6

Given a function . Investigate the function for continuity at points. Build a graph.

Solution: and again immediately execute the drawing on the draft:

The peculiarity of this graph is that the piecewise function is given by the equation of the abscissa axis. Here this area is drawn in green, but in a notebook it is usually highlighted in bold with a simple pencil. And, of course, don’t forget about our rams: the value belongs to the tangent branch (red dot), and the value belongs to the straight line.

Everything is clear from the drawing - the function is continuous along the entire number line, all that remains is to formalize the solution, which is brought to full automation literally after 3-4 similar examples:

I) We examine the point for continuity

1) – the function is defined at a given point.

2) Let's calculate one-sided limits:

, which means there is a general limit.

Just in case, let me remind you of a trivial fact: the limit of a constant is equal to the constant itself. In this case, the limit of zero is equal to zero itself (left-handed limit).

3) – the limit of a function at a point is equal to the value of this function at a given point.

Thus, a function is continuous at a point by the definition of continuity of a function at a point.

II) We examine the point for continuity

1) – the function is defined at a given point.

2) Find one-sided limits:

And here – the limit of one is equal to the unit itself.

– there is a general limit.

3) – the limit of a function at a point is equal to the value of this function at a given point.

Thus, a function is continuous at a point by the definition of continuity of a function at a point.

As usual, after research we transfer our drawing to the final version.

Answer: the function is continuous at the points.

Please note that in the condition we were not asked anything about studying the entire function for continuity, and it is considered good mathematical form to formulate precise and clear the answer to the question posed. By the way, if the conditions do not require you to build a graph, then you have every right not to build it (although later the teacher can force you to do this).

A small mathematical “tongue twister” for solving it yourself:

Example 7

Given a function . Investigate the function for continuity at points. Classify breakpoints, if any. Execute the drawing.

Try to “pronounce” all the “words” correctly =) And draw the graph more precisely, accuracy, it will not be superfluous everywhere;-)

As you remember, I recommended immediately completing the drawing as a draft, but from time to time you come across examples where you can’t immediately figure out what the graph looks like. Therefore, in some cases, it is advantageous to first find one-sided limits and only then, based on the study, depict the branches. In the final two examples we will also learn a technique for calculating some one-sided limits:

Example 8

Examine the function for continuity and construct its schematic graph.

Solution: the bad points are obvious: (reduces the denominator of the exponent to zero) and (reduces the denominator of the entire fraction to zero). It is not clear what the graph of this function looks like, which means it is better to do some research first.

Definitions and formulations of the main theorems and properties of a continuous function of one variable are given. The properties of a continuous function at a point, on a segment, the limit and continuity of a complex function, and the classification of discontinuity points are considered. Definitions and theorems related to the inverse function are given. The properties of elementary functions are outlined.

Content

We can formulate the concept of continuity in in terms of increments. To do this, we introduce a new variable, which is called the increment of the variable x at the point. Then the function is continuous at the point if
.
Let's introduce a new function:
.
They call her function increment at point . Then the function is continuous at the point if
.

Definition of continuity on the right (left)
Function f (x) called continuous on the right (left) at point x 0 , if it is defined on some right-sided (left-sided) neighborhood of this point, and if the right (left) limit at the point x 0 equal to the function value at x 0 :
.

Theorem on the boundedness of a continuous function
Let the function f (x) is continuous at point x 0 . Then there is a neighborhood U (x0), on which the function is limited.

Theorem on the preservation of the sign of a continuous function
Let the function be continuous at the point. And let it have a positive (negative) value at this point:
.
Then there is a neighborhood of the point where the function has a positive (negative) value:
at .

Arithmetic properties of continuous functions
Let the functions and be continuous at the point .
Then the functions , and are continuous at the point .
If , then the function is continuous at the point .

Left-right continuity property
A function is continuous at a point if and only if it is continuous on the right and left.

Proofs of the properties are given on the page “Properties of functions continuous at a point”.

Continuity of a complex function

Continuity theorem for a complex function
Let the function be continuous at the point. And let the function be continuous at the point.
Then the complex function is continuous at the point.

Limit of a complex function

Theorem on the limit of a continuous function of a function
Let there be a limit of the function at , and it is equal to:
.
Here is point t 0 can be finite or infinitely distant: .
And let the function be continuous at the point.
Then there is a limit of a complex function, and it is equal to:
.

Theorem on the limit of a complex function
Let the function have a limit and map a punctured neighborhood of a point onto a punctured neighborhood of a point. Let the function be defined on this neighborhood and have a limit on it.
Here are the final or infinitely distant points: . Neighborhoods and their corresponding limits can be either two-sided or one-sided.
Then there is a limit of a complex function and it is equal to:
.

Break points

Determining the break point
Let the function be defined on some punctured neighborhood of the point . The point is called function break point, if one of two conditions is met:
1) not defined in ;
2) is defined at , but is not at this point.

Determination of the discontinuity point of the 1st kind
The point is called discontinuity point of the first kind, if is a break point and there are finite one-sided limits on the left and right:
.

Definition of a function jump
Jump Δ function at a point is the difference between the limits on the right and left
.

Determining the break point
The point is called removable break point, if there is a limit
,
but the function at the point is either not defined or is not equal to the limit value: .

Thus, the point of removable discontinuity is the point of discontinuity of the 1st kind, at which the jump of the function is equal to zero.

Determination of the discontinuity point of the 2nd kind
The point is called point of discontinuity of the second kind, if it is not a discontinuity point of the 1st kind. That is, if there is not at least one one-sided limit, or at least one one-sided limit at a point is equal to infinity.

Properties of functions continuous on an interval

Definition of a function continuous on an interval
A function is called continuous on an interval (at) if it is continuous at all points of the open interval (at) and at points a and b, respectively.

Weierstrass's first theorem on the boundedness of a function continuous on an interval
If a function is continuous on an interval, then it is bounded on this interval.

Determining the attainability of the maximum (minimum)
A function reaches its maximum (minimum) on the set if there is an argument for which
for all .

Determining the reachability of the upper (lower) face
A function reaches its upper (lower) bound on the set if there is an argument for which
.

Weierstrass's second theorem on the maximum and minimum of a continuous function
A function continuous on a segment reaches its upper and lower bounds on it or, which is the same, reaches its maximum and minimum on the segment.

Bolzano-Cauchy intermediate value theorem
Let the function be continuous on the segment. And let C be an arbitrary number located between the values of the function at the ends of the segment: and . Then there is a point for which
.

Corollary 1
Let the function be continuous on the segment. And let the function values at the ends of the segment have different signs: or . Then there is a point at which the value of the function is equal to zero:
.

Corollary 2
Let the function be continuous on the segment. Let it go . Then the function takes on the interval all the values from and only these values:
at .

Inverse functions

Definition of an inverse function
Let a function have a domain of definition X and a set of values Y. And let it have the property:
for all .
Then for any element from the set Y one can associate only one element of the set X for which . This correspondence defines a function called inverse function To . The inverse function is denoted as follows:
.

From the definition it follows that
;
for all ;
for all .

Lemma on the mutual monotonicity of direct and inverse functions
If a function is strictly increasing (decreasing), then there is an inverse function that is also strictly increasing (decreasing).

Property of symmetry of graphs of direct and inverse functions
The graphs of direct and inverse functions are symmetrical with respect to the straight line.

Theorem on the existence and continuity of an inverse function on an interval
Let the function be continuous and strictly increasing (decreasing) on the segment. Then the inverse function is defined and continuous on the segment, which strictly increases (decreases).

For an increasing function. For decreasing - .

Theorem on the existence and continuity of an inverse function on an interval
Let the function be continuous and strictly increasing (decreasing) on an open finite or infinite interval. Then the inverse function is defined and continuous on the interval, which strictly increases (decreases).

For an increasing function.
For decreasing: .

In a similar way, we can formulate the theorem on the existence and continuity of the inverse function on a half-interval.

Properties and continuity of elementary functions

Elementary functions and their inverses are continuous in their domain of definition. Below we present the formulations of the corresponding theorems and provide links to their proofs.

Exponential function

Exponential function f (x) = a x, with base a > 0 is the limit of the sequence
,
where is an arbitrary sequence of rational numbers tending to x:
.

Theorem. Properties of the Exponential Function
The exponential function has the following properties:
(P.0) defined, for , for all ;
(P.1) for a ≠ 1 has many meanings;
(P.2) strictly increases at , strictly decreases at , is constant at ;
(P.3) ;
(P.3*) ;
(P.4) ;
(P.5) ;
(P.6) ;
(P.7) ;
(P.8) continuous for all;
(P.9) at ;
at .

Logarithm

Logarithmic function, or logarithm, y = log a x, with base a is the inverse of the exponential function with base a.

Theorem. Properties of the logarithm
Logarithmic function with base a, y = log a x, has the following properties:
(L.1) defined and continuous, for and , for positive values of the argument;
(L.2) has many meanings;
(L.3) strictly increases as , strictly decreases as ;
(L.4) at ;
at ;
(L.5) ;
(L.6) at ;
(L.7) at ;
(L.8) at ;
(L.9) at .

Exponent and natural logarithm

In the definitions of the exponential function and the logarithm, a constant appears, which is called the base of the power or the base of the logarithm. In mathematical analysis, in the vast majority of cases, simpler calculations are obtained if the number e is used as the basis:
.
An exponential function with base e is called an exponent: , and a logarithm with base e is called a natural logarithm: .

The properties of the exponent and the natural logarithm are presented on the pages
"Exponent, e to the power of x",
"Natural logarithm, ln x function"

Power function

Power function with exponent p is the function f (x) = x p, the value of which at point x is equal to the value of the exponential function with base x at point p.
In addition, f (0) = 0 p = 0 for p > 0 .

Here we will consider the properties of the power function y = x p for non-negative values of the argument. For rationals, for odd m, the power function is also defined for negative x. In this case, its properties can be obtained using even or odd.
These cases are discussed in detail and illustrated on the page “Power function, its properties and graphs”.

Theorem. Properties of the power function (x ≥ 0)
A power function, y = x p, with exponent p has the following properties:
(C.1) defined and continuous on the set
at ,
at ".

Trigonometric functions

Theorem on the continuity of trigonometric functions
Trigonometric functions: sine ( sin x), cosine ( cos x), tangent ( tg x) and cotangent ( ctg x

Theorem on the continuity of inverse trigonometric functions
Inverse trigonometric functions: arcsine ( arcsin x), arc cosine ( arccos x), arctangent ( arctan x) and arc tangent ( arcctg x), are continuous in their domains of definition.

References:
O.I. Besov. Lectures on mathematical analysis. Part 1. Moscow, 2004.
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.

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✪ Continuity of function and breakpoints of function

✪ 15 Continuous function

✪ Continuous features

✪ Mathematical analysis, lesson 5, Continuity of function

✪ Continuous random variable. Distribution function

Subtitles

Definition

If you “correct” the function f (\displaystyle f) at the point of removable rupture and put f (a) = lim x → a f (x) (\displaystyle f(a)=\lim \limits _(x\to a)f(x)), then we get a function that is continuous at a given point. This operation on a function is called extending the function to continuous or redefinition of the function by continuity, which justifies the name of the point as a point removable rupture.

Break point "jump"

A “jump” discontinuity occurs if

lim x → a − 0 f (x) ≠ lim x → a + 0 f (x) (\displaystyle \lim \limits _(x\to a-0)f(x)\neq \lim \limits _(x \to a+0)f(x)).

Break point "pole"

A pole gap occurs if one of the one-sided limits is infinite.

lim x → a − 0 f (x) = ± ∞ (\displaystyle \lim \limits _(x\to a-0)f(x)=\pm \infty ) or lim x → a + 0 f (x) = ± ∞ (\displaystyle \lim \limits _(x\to a+0)f(x)=\pm \infty ). [ ]

Significant break point

At the point of significant discontinuity, one of the one-sided limits is completely absent.

Classification of isolated singular points in Rn, n>1

For functions f: R n → R n (\displaystyle f:\mathbb (R) ^(n)\to \mathbb (R) ^(n)) And f: C → C (\displaystyle f:\mathbb (C) \to \mathbb (C) ) There is no need to work with break points, but often you have to work with singular points (points where the function is not defined). The classification is similar.

The concept of “leap” is missing. What's in R (\displaystyle \mathbb (R) ) is considered a jump; in spaces of higher dimensions it is an essential singular point.

Properties

Local

Function continuous at a point a (\displaystyle a), is bounded in some neighborhood of this point.
If the function f (\displaystyle f) continuous at a point a (\displaystyle a) And f (a) > 0 (\displaystyle f(a)>0)(or f(a)< 0 {\displaystyle f(a)<0} ), That f (x) > 0 (\displaystyle f(x)>0)(or f(x)< 0 {\displaystyle f(x)<0} ) for all x (\displaystyle x), quite close to a (\displaystyle a).
If the functions f (\displaystyle f) And g (\displaystyle g) continuous at a point a (\displaystyle a), then the functions f + g (\displaystyle f+g) And f ⋅ g (\displaystyle f\cdot g) are also continuous at a point a (\displaystyle a).
If the functions f (\displaystyle f) And g (\displaystyle g) continuous at a point a (\displaystyle a) and wherein g (a) ≠ 0 (\displaystyle g(a)\neq 0), then the function f / g (\displaystyle f/g) is also continuous at a point a (\displaystyle a).
If the function f (\displaystyle f) continuous at a point a (\displaystyle a) and function g (\displaystyle g) continuous at a point b = f (a) (\displaystyle b=f(a)), then their composition h = g ∘ f (\displaystyle h=g\circ f) continuous at a point a (\displaystyle a).

Global

compact set) is uniformly continuous on it.
A function that is continuous on a segment (or any other compact set) is bounded and reaches its maximum and minimum values on it.
Function range f (\displaystyle f), continuous on the segment , is the segment [ min f , max f ] , (\displaystyle [\min f,\ \max f],) where the minimum and maximum are taken along the segment [ a , b ] (\displaystyle ).
If the function f (\displaystyle f) continuous on the segment [ a , b ] (\displaystyle ) And f (a) ⋅ f (b)< 0 , {\displaystyle f(a)\cdot f(b)<0,} then there is a point at which f (ξ) = 0 (\displaystyle f(\xi)=0).
If the function f (\displaystyle f) continuous on the segment [ a , b ] (\displaystyle ) and number φ (\displaystyle \varphi ) satisfies the inequality f(a)< φ < f (b) {\displaystyle f(a)<\varphi or inequality f (a) > φ > f (b) , (\displaystyle f(a)>\varphi >f(b),) then there is a point ξ ∈ (a , b) , (\displaystyle \xi \in (a,b),) wherein f (ξ) = φ (\displaystyle f(\xi)=\varphi ).
A continuous mapping of a segment to the real line is injective if and only if the given function on the segment is strictly monotonic.
Monotonic function on a segment [ a , b ] (\displaystyle ) is continuous if and only if its range of values is a segment with ends f (a) (\displaystyle f(a)) And f (b) (\displaystyle f(b)).
If the functions f (\displaystyle f) And g (\displaystyle g) continuous on the segment [ a , b ] (\displaystyle ), and f(a)< g (a) {\displaystyle f(a) And f (b) > g (b) , (\displaystyle f(b)>g(b),) then there is a point ξ ∈ (a , b) , (\displaystyle \xi \in (a,b),) wherein f (ξ) = g (ξ) . (\displaystyle f(\xi)=g(\xi).) From here, in particular, it follows that any continuous mapping of a segment into itself has at least one fixed point.

Examples

Elementary functions

This function is continuous at every point x ≠ 0 (\displaystyle x\neq 0).

The point is the break point first kind, and

lim x → 0 − f (x) = − 1 ≠ 1 = lim x → 0 + f (x) (\displaystyle \lim \limits _(x\to 0-)f(x)=-1\neq 1= \lim \limits _(x\to 0+)f(x)),

while at the point itself the function vanishes.

Step function

Step function defined as

f (x) = ( 1 , x ⩾ 0 0 , x< 0 , x ∈ R {\displaystyle f(x)={\begin{cases}1,&x\geqslant 0\\0,&x<0\end{cases}},\quad x\in \mathbb {R} }

is continuous everywhere except the point x = 0 (\displaystyle x=0), where the function suffers a discontinuity of the first kind. However, at the point x = 0 (\displaystyle x=0) there is a right-hand limit that coincides with the value of the function at a given point. So this function is an example continuous on the right functions throughout the entire definition area.

Similarly, the step function defined as

f (x) = ( 1 , x > 0 0 , x ⩽ 0 , x ∈ R (\displaystyle f(x)=(\begin(cases)1,&x>0\\0,&x\leqslant 0\end( cases)),\quad x\in \mathbb (R) )

is an example continuous on the left functions throughout the entire definition area.

Dirichlet function

f (x) = ( 1 , x ∈ Q 0 , x ∈ R ∖ Q (\displaystyle f(x)=(\begin(cases)1,&x\in \mathbb (Q) \\0,&x\in \ mathbb (R) \setminus \mathbb (Q) \end(cases)))

The study of a function for continuity at a point is carried out according to an already established routine scheme, which consists of checking three conditions of continuity:

Example 1

Examine the function for continuity. Determine the nature of the function discontinuities, if they exist. Execute the drawing.

Solution:

1) The only point within the scope is where the function is not defined.

One-sided limits are finite and equal.

Thus, at the point the function suffers a removable discontinuity.

What does the graph of this function look like?

I would like to simplify , and it seems like an ordinary parabola is obtained. BUT the original function is not defined at point , so the following clause is required:

Let's make the drawing:

Answer: the function is continuous on the entire number line except the point at which it suffers a removable discontinuity.

The function can be further defined in a good or not so good way, but according to the condition this is not required.

You say this is a far-fetched example? Not at all. This has happened dozens of times in practice. Almost all of the site’s tasks come from real independent work and tests.

Let's get rid of our favorite modules:

Example 2

Explore function for continuity. Determine the nature of the function discontinuities, if they exist. Execute the drawing.

Let us examine the function for continuity analytically:

1) The function is not defined at the point, so we can immediately say that it is not continuous at it.

2) Let’s establish the nature of the discontinuity; to do this, we calculate one-sided limits:

The one-sided limits are finite and different, which means that the function suffers a discontinuity of the 1st kind with a jump at the point . Note that it doesn't matter whether the function at the break point is defined or not.

Now all that remains is to transfer the drawing from the draft (it was made as if with the help of research ;-)) and complete the task:

Answer: the function is continuous on the entire number line except for the point at which it suffers a discontinuity of the first kind with a jump.

Example 3

Explore function for continuity. Determine the nature of the function discontinuities, if they exist. Make a drawing.

This is an example for you to solve on your own, a sample solution at the end of the lesson.

Let's move on to the most popular and widespread version of the task, when the function consists of three parts:

Example 4

Examine a function for continuity and plot a graph of the function

The one-sided limits are finite and different, which means that the function suffers a discontinuity of the 1st kind with a jump at the point .

Let us calculate the discontinuity jump as the difference between the right and left limits:
, that is, the graph jerked up one unit.

II) We examine the point for continuity

1) - the function is defined at a given point.

2) Find one-sided limits:

- one-sided limits are finite and equal, which means there is a general limit.

At the final stage, we transfer the drawing to the final version, after which we put the final chord:

Answer: the function is continuous on the entire number line, except for the point at which it suffers a discontinuity of the first kind with a jump.

Example 5

Examine a function for continuity and construct its graph .

This is an example for independent solution, a short solution and an approximate sample of the problem at the end of the lesson.

Example 6

Given a function . Investigate the function for continuity at points. Build a graph.

Solution: and again immediately execute the drawing on the draft:

I) We examine the point for continuity

2) Let's calculate one-sided limits:

, which means there is a general limit.

A little funny thing happened here. The fact is that I created a lot of materials about the limits of a function, and several times I wanted to, but several times I forgot about one simple question. And so, with an incredible effort of will, I forced myself not to lose the thought =) Most likely, some “dummies” readers doubt: what is the limit of the constant? The limit of a constant is equal to the constant itself. In this case, the limit of zero is equal to zero itself (left-handed limit).

3) - the limit of a function at a point is equal to the value of this function at a given point.

Thus, a function is continuous at a point by the definition of continuity of a function at a point.

II) We examine the point for continuity

1) - the function is defined at a given point.

2) Find one-sided limits:

And here, in the right-hand limit, the limit of unity is equal to unity itself.

- there is a general limit.

3) - the limit of a function at a point is equal to the value of this function at a given point.

Thus, a function is continuous at a point by the definition of continuity of a function at a point.

As usual, after research we transfer our drawing to the final version.

Answer: the function is continuous at the points.

A small mathematical “tongue twister” for solving it yourself:

Example 7

Given a function .

Investigate the function for continuity at points. Classify breakpoints, if any. Execute the drawing.

Try to “pronounce” all the “words” correctly =) And draw the graph more precisely, accuracy, it will not be superfluous everywhere;-)

Example 8

Examine the function for continuity and construct its schematic graph.

Solution: the bad points are obvious: (reduces the denominator of the exponent to zero) and (reduces the denominator of the entire fraction to zero). It’s not clear what the graph of this function looks like, which means it’s better to do some research first:

I) We examine the point for continuity

2) Find one-sided limits:

pay attention to typical method for calculating a one-sided limit: instead of “x” we substitute . There is no crime in the denominator: the “addition” “minus zero” does not play a role, and the result is “four”. But in the numerator there is a little thriller going on: first we kill -1 and 1 in the denominator of the indicator, resulting in . Unit divided by , is equal to “minus infinity”, therefore: . And finally, the “two” in infinitely large negative degree equal to zero: . Or, to be even more specific: .

Let's calculate the right-hand limit:

And here - instead of “X” we substitute . In the denominator, the “additive” again does not play a role: . In the numerator, actions similar to the previous limit are carried out: we destroy opposite numbers and divide one by :

The right-hand limit is infinite, which means that the function suffers a discontinuity of the 2nd kind at the point .

II) We examine the point for continuity

1) The function is not defined at this point.

2) Let's calculate the left-sided limit:

The method is the same: we substitute “X” into the function. There is nothing interesting in the numerator - it turns out to be a finite positive number. And in the denominator we open the brackets, remove the “threes”, and the “additive” plays a decisive role.

As a result, the final positive number divided by infinitesimal positive number, gives “plus infinity”: .

The right-hand limit is like a twin brother, with the only exception that it appears in the denominator infinitesimal negative number:

One-sided limits are infinite, which means that the function suffers a discontinuity of the 2nd kind at the point .

Thus, we have two break points, and, obviously, three branches of the graph. For each branch, it is advisable to carry out a point-by-point construction, i.e. take several “x” values and substitute them into . Please note that the condition allows for the construction of a schematic drawing, and such relaxation is natural for manual work. I build graphs using a program, so I don’t have such difficulties, here’s a fairly accurate picture:

Direct are vertical asymptotes for the graph of this function.

Answer: the function is continuous on the entire number line except for points at which it suffers discontinuities of the 2nd kind.

A simpler function to solve on your own:

Example 9

Examine the function for continuity and make a schematic drawing.

An approximate example of a solution at the end that crept up unnoticed.

See you soon!

Solutions and answers:

Example 3:Solution : transform the function: . Considering the modulus disclosure rule and the fact that , we rewrite the function in piecewise form:

Let's examine the function for continuity.

1) The function is not defined at the point .

The one-sided limits are finite and different, which means that the function suffers a discontinuity of the 1st kind with a jump at the point . Let's make the drawing:

Answer: the function is continuous on the entire number line except the point , in which it suffers a discontinuity of the first kind with a jump. Jump Gap: (two units up).

Example 5:Solution : Each of the three parts of the function is continuous on its own interval.
I)
1)

2) Let's calculate one-sided limits:

, which means there is a general limit.
3) - the limit of a function at a point is equal to the value of this function at a given point.
So the function continuous at a point by defining the continuity of a function at a point.
II) We examine the point for continuity

1) - the function is defined at a given point. the function suffers a discontinuity of the 2nd kind at the point

How to find the domain of a function?

Examples of solutions

If something is missing somewhere, it means there is something somewhere

We continue to study the “Functions and Graphs” section, and the next station on our journey is Function Domain. An active discussion of this concept began in the first lesson. about function graphs, where I looked at elementary functions, and, in particular, their domains of definition. Therefore, I recommend that dummies start with the basics of the topic, since I will not dwell on some basic points again.

It is assumed that the reader knows the domains of definition of the basic functions: linear, quadratic, cubic functions, polynomials, exponential, logarithm, sine, cosine. They are defined on . For tangents, arcsines, so be it, I forgive you =) Rarer graphs are not immediately remembered.

The scope of definition seems to be a simple thing, and a logical question arises: what will the article be about? In this lesson I will look at common problems of finding the domain of a function. Moreover, we will repeat inequalities with one variable, the solution skills of which will also be required in other problems of higher mathematics. The material, by the way, is all school material, so it will be useful not only for students, but also for students. The information, of course, does not pretend to be encyclopedic, but here are not far-fetched “dead” examples, but roasted chestnuts, which are taken from real practical works.

Let's start with a quick dive into the topic. Briefly about the main thing: we are talking about a function of one variable. Its domain of definition is many meanings of "x", for which exist meanings of "players". Let's look at a hypothetical example:

The domain of definition of this function is a union of intervals:
(for those who forgot: - unification icon). In other words, if you take any value of “x” from the interval , or from , or from , then for each such “x” there will be a value “y”.

Roughly speaking, where the domain of definition is, there is a graph of the function. But the half-interval and the “tse” point are not included in the definition area, so there is no graph there.

Yes, by the way, if anything is not clear from the terminology and/or content of the first paragraphs, it is better to return to the article Graphs and properties of elementary functions.