Three definitions of continuity of a function at a point. Properties of functions continuous on an interval

A continuous function is a function without “jumps”, that is, one for which the condition is satisfied: small changes in the argument are followed by small changes in the corresponding values of the function. The graph of such a function is a smooth or continuous curve.

Continuity at a limit point for a certain set can be defined using the concept of a limit, namely: a function must have a limit at this point that is equal to its value at the limit point.

If these conditions are violated at a certain point, they say that the function at this point suffers a discontinuity, that is, its continuity is violated. In the language of limits, a break point can be described as a discrepancy between the value of a function at the break point and the limit of the function (if it exists).

The break point can be removable; for this, the existence of a limit of the function is necessary, but it does not coincide with its value at a given point. In this case, it can be “corrected” at this point, that is, it can be further defined to continuity.
A completely different picture emerges if there is a limit to the given function. There are two possible breakpoint options:

of the first kind - both of the one-sided limits are available and finite, and the value of one of them or both does not coincide with the value of the function at a given point;
of the second kind, when one or both of the one-sided limits do not exist or their values are infinite.

Properties of continuous functions

The function obtained as a result of arithmetic operations, as well as the superposition of continuous functions on their domain of definition, is also continuous.
If you are given a continuous function that is positive at some point, then you can always find a sufficiently small neighborhood of it where it will retain its sign.
Similarly, if its values at two points A and B are equal to a and b, respectively, and a is different from b, then for intermediate points it will take all values from the interval (a ; b). From this we can draw an interesting conclusion: if you let a stretched elastic band compress so that it does not sag (remains straight), then one of its points will remain motionless. And geometrically, this means that there is a straight line passing through any intermediate point between A and B that intersects the graph of the function.

Let us note some of the continuous (in the domain of their definition) elementary functions:

constant;
rational;
trigonometric.

There is an inextricable connection between two fundamental concepts in mathematics - continuity and differentiability. It is enough just to remember that for a function to be differentiable it is necessary that it be a continuous function.

If a function is differentiable at some point, then it is continuous there. However, it is not at all necessary that its derivative be continuous.

A function that has a continuous derivative on a certain set belongs to a separate class of smooth functions. In other words, it is a continuously differentiable function. If the derivative has a limited number of discontinuity points (only of the first kind), then such a function is called piecewise smooth.

Another important concept is the uniform continuity of a function, that is, its ability to be equally continuous at any point in its domain of definition. Thus, this is a property that is considered at many points, and not at any one point.

If we fix a point, then we get nothing more than a definition of continuity, that is, from the presence of uniform continuity it follows that we have a continuous function. Generally speaking, the converse is not true. However, according to Cantor’s theorem, if a function is continuous on a compact set, that is, on a closed interval, then it is uniformly continuous on it.

Determining the continuity of a function at a point
Function f (x) called continuous at point x 0 neighborhood U (x0) this point, and if the limit as x tends to x 0 exists and is equal to the value of the function at x 0 :
.

This implies that x 0 - this is the end point. The function value in it can only be a finite number.

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 :
.

Examples

Example 1

Using the Heine and Cauchy definitions, prove that the function is continuous for all x.

Let there be an arbitrary number. Let us prove that the given function is continuous at the point. The function is defined for all x . Therefore, it is defined at a point and in any of its neighborhoods.

We use Heine's definition

Let's use . Let there be an arbitrary sequence converging to : . Applying the property of the limit of a product of sequences we have:
.
Since there is an arbitrary sequence converging to , then
.
Continuity has been proven.

We use the Cauchy definition

Let's use .
Let's consider the case. We have the right to consider the function on any neighborhood of the point. Therefore we will assume that
(A1.1) .

Let's apply the formula:
.
Taking into account (A1.1), we make the following estimate:

;
(A1.2) .

Applying (A1.2), we estimate the absolute value of the difference:
;
(A1.3) .
.
According to the properties of inequalities, if (A1.3) is satisfied, if and if , then .

Now let's look at the point. In this case
.
.

.
This means that the function is continuous at the point.

In a similar way, one can prove that the function , where n is a natural number, is continuous on the entire real axis.

Example 2

Using prove that the function is continuous for all .

The given function is defined at . Let us prove that it is continuous at the point.

Let's consider the case.
We have the right to consider the function on any neighborhood of the point. Therefore we will assume that
(A2.1) .

Let's apply the formula:
(A2.2) .
Let's put it. Then
.

Taking into account (A2.1), we make the following estimate:

.
So,
.

Applying this inequality and using (A2.2), we estimate the difference:

.
So,
(A2.3) .

We introduce positive numbers and , connecting them with the following relations:
.
According to the properties of inequalities, if (A2.3) is satisfied, if and if , then .

This means that for any positive there is always a . Then for all x satisfying the inequality, the following inequality is automatically satisfied:
.
This means that the function is continuous at the point.

Now let's look at the point. We need to show that the given function is continuous at this point on the right. In this case
.
Enter positive numbers and :
.

This shows that for any positive there is always . Then for all x such that , the following inequality holds:
.
It means that . That is, the function is continuous on the right at the point.

In a similar way, one can prove that the function , where n is a natural number, is continuous for .

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.

1. Introduction.

2. Determination of continuity of a function.

3. Classification of break points

4. Properties of continuous functions.

5. The economic meaning of continuity.

6. Conclusion.

10.1. Introduction

Whenever we assess the inevitable changes in the world around us over time, we try to analyze the ongoing processes in order to highlight their most significant features. One of the first questions to arise along this path is: How changes characteristic of this phenomenon occur - continuously or discretely, i.e. spasmodically. Is the currency rate depreciating or collapsing evenly, is there a gradual evolution or a revolutionary leap? In order to unify qualitative and quantitative assessments of what is happening, one should abstract from the specific content and study the problem in terms of functional dependence. This can be done by the theory of limits, which we discussed in the last lecture.

10.2. Definition of continuity of a function

The continuity of a function is intuitively related to the fact that its graph is a continuous curve that does not break anywhere. We draw a graph of such a function without lifting our pen from the paper. If a function is given in a table, then, strictly speaking, its continuity cannot be judged, because for a given table step the behavior of the function in intervals is not defined.

In reality, with continuity, the following circumstance occurs: if the parameters characterizing the situation A little change then A little the situation will change. The important thing here is not that the situation will change, but that it will change “a little.”

Let us formulate the concept of continuity in the language of increments. Let some phenomenon be described by a function and point a belongs to the domain of definition of the function. The difference is called argument increment at the point a, difference – function increment at the point a.

Definition 10.1.Function continuous at a point a, if it is defined at this point and an infinitesimal increment in the argument corresponds to an infinitesimal increment in the function:

Example 10.1. Examine the continuity of the function at the point.

Solution. Let's build a graph of the function and mark the increments D on it x and D y(Fig. 10.1).

The graph shows that the smaller the increment D x, the less D y. Let's show this analytically. The increment of the argument is equal to , then the increment of the function at this point will be equal to

From this it is clear that if , then and:

Let us give another definition of the continuity of a function.

Definition 10.2.The function is called continuous at point a if:

1) it is defined at point a and some of its surroundings;

2) one-sided limits exist and are equal to each other:

;

3) limit of the function at x® a is equal to the value of the function at this point:

If at least one of these conditions is violated, then the function is said to undergo gap.

This definition is operational for establishing continuity at a point. Following his algorithm and noting the coincidences and discrepancies between the requirements of the definition and a specific example, we can conclude that the function is continuous at a point.

In Definition 2, the idea of proximity clearly emerges when we introduced the concept of limit. With an unlimited approximation of the argument x to the limit value a, continuous at a point a function f(x) approaches the limiting value arbitrarily close f(a).

10.3. Classification of break points

The points at which the continuity conditions of a function are violated are called break points this function. If x 0 is the break point of the function; at least one of the conditions for the continuity of the function is not satisfied. Consider the following example.

1. The function is defined in a certain neighborhood of the point a, but not defined at the point itself a. For example, the function is not defined at point a=2, therefore undergoes a discontinuity (see Fig. 10.2).

Rice. 10.2 Fig. 10.3

2. The function is defined at a point a and in some of its neighborhood, its one-sided limits exist, but are not equal to each other: , then the function undergoes a discontinuity. For example, the function

is defined at the point, but at the function experiences a discontinuity (see Fig. 10.3), because

And ().

3. The function is defined at a point a and in some neighborhood of it, there is a limit of the function at , but this limit is not equal to the value of the function at the point a:

For example, the function (see Fig. 10.4)

Here is the breaking point:

All discontinuity points are divided into removable discontinuity points, discontinuity points of the first and second kind.

Definition 10.1. The break point is called the point repairable gap , if at this point there are finite limits of the function on the left and on the right, equal to each other:

The limit of the function at this point exists, but is not equal to the value of the function at the limit point (if the function is defined at the limit point), or the function at the limit point is not defined.

In Fig. 10.4 at the point the continuity conditions are violated, and the function has a discontinuity. Point on the graph (0; 1) gouged out. However, this gap can be easily eliminated - it is enough to redefine this function, setting it equal to its limit at this point, i.e. put . Therefore, such gaps are called removable.

Definition 10.2. The breaking point is called discontinuity point of the 1st kind , if at this point there are finite limits of the function on the left and on the right, but they are not equal to each other:

At this point the function is said to experience leap.

In Fig. 10.3 the function has a discontinuity of the 1st kind at the point. The left and right limits at this point are equal:

And .

The jump of the function at the discontinuity point is equal to .

It is impossible to define such a function as continuous. The graph consists of two half-lines separated by a jump.

Definition 10.3. The breaking point is called discontinuity point of the 2nd kind , if at least one of the one-sided limits of the function (left or right) does not exist or is equal to infinity.

In Figure 10.3, the function at a point has a discontinuity of the 2nd kind. The considered function at is infinitely large and has no finite limit either on the right or on the left. Therefore, there is no need to talk about continuity at such a point.

Example 10.2. Construct a graph and determine the nature of the break points:

Solution. Let's plot the function f(x) (Figure 10.5).

The figure shows that the original function has three discontinuity points: , x 2 = 1,
x 3 = 3. Let's consider them in order.

Therefore the point has rupture of the 2nd kind.

a) The function is defined at this point: f(1) = –1.

b) , ,

those. at the point x 2 = 1 available repairable gap. By redefining the function value at this point: f(1) = 5, the discontinuity is eliminated and the function at this point becomes continuous.

a) The function is defined at this point: f(3) = 1.

So, at the point x 1 = 3 available rupture of the 1st kind. The function at this point experiences a jump equal to D y= –2–1 = –3.

10.4. Properties of continuous functions

Recalling the corresponding properties of limits, we conclude that functions that are the result of arithmetic operations on functions continuous at the same point are also continuous. Note:

1) if the functions and are continuous at the point a, then the functions , and (provided that ) are also continuous at this point;

2) if the function is continuous at the point a and the function is continuous at the point , then the complex function is continuous at the point a And

those. the limit sign can be placed under the sign of a continuous function.

They say that a function is continuous on some set if it is continuous at every point of this set. The graph of such a function is a continuous line that can be crossed out with one stroke of the pen.

All major elementary functions are continuous at all points where they are defined.

Functions, continuous on the segment, have a number of important distinctive properties. Let us formulate theorems expressing some of these properties.

Theorem 10.1 (Weierstrass's theorem ). If a function is continuous on a segment, then it reaches its minimum and maximum values on this segment.

Theorem 10.2 (Cauchy's theorem ). If a function is continuous on an interval, then on this interval all intermediate values between the smallest and largest values.

The following important property follows from Cauchy's theorem.

Theorem 10.3. If a function is continuous on a segment and takes on values of different signs at the ends of the segment, then between a and b there is a point c at which the function vanishes:.

The geometric meaning of this theorem is obvious: if the graph of a continuous function goes from the lower half-plane to the upper half-plane (or vice versa), then at least at one point it will intersect the axis Ox(Fig. 10.6).

Example 10.3. Approximately calculate the root of the equation

, (i.e. approximately replace) polynomial of the corresponding degree.

This is a very important property of continuous functions for practice. For example, very often continuous functions are specified by tables (observational or experimental data). Then, using some method, you can replace the tabulated function with a polynomial. In accordance with Theorem 10.3, this can always be done with sufficiently high accuracy. Working with an analytically defined function (especially a polynomial) is much easier.

10.5. Economic meaning of continuity

Most of the functions used in economics are continuous, and this allows one to make quite significant statements of economic content.

To illustrate, consider the following example.

Tax rate N has approximately the same graph as in Fig. 10.7a.

At the ends of the intervals it is discontinuous and these discontinuities are of the 1st kind. However, the amount of income tax itself P(Fig. 10.7b) is a continuous function of annual income Q. From here, in particular, it follows that if the annual incomes of two people differ insignificantly, then the difference in the amounts of income tax that they must pay should also differ insignificantly. It is interesting that the circumstance is perceived by the vast majority of people as completely natural, which they do not even think about.

10.6. Conclusion

Toward the end, let’s allow ourselves a small retreat.

Here's how to graphically express the sad observation of the ancients:

Sic transit Gloria mundi...

(This is how earthly glory passes …)

End of work -

This topic belongs to the section:

Concept of function

The concept of function.. everything flows and everything changes Heraclitus.. table x x x x y y y y y..

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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.

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 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.

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:

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.

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

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.

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’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.

Definition
Function f (x) called continuous at point x 0 neighborhood of this point, and if the limit as x tends to x 0 equal to the function value at x 0 :
.

Using the Cauchy and Heine definitions of the limit of a function, we can give expanded definitions of the continuity of a function at a point .

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
.

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.