Example 1. Prove that the function y=sinx is continuous for any value of x.
Solution. Let x 0 be an arbitrary point. Giving it an increment Δx, we get the point x=x 0 +Δx. Then . We get .
Definition. The function y=f(x) is called continuous at the point x 0 on the right (left) if
.
A function continuous at an interior point will be both right and left continuous. The converse is also true: if a function is continuous at a point on the left and right, then it will be continuous at that point. However, a function can only be continuous on one side. For example, for , , f(1)=1, therefore, this function is continuous only on the left (for the graph of this function, see above in paragraph 5.7.2).
Definition. A function is called continuous on some interval if it is continuous at every point of this interval.
In particular, if the interval is a segment, then one-sided continuity is implied at its ends.

Properties of continuous functions

Function breaks and their classification

The following classification is adopted for break points.
1) If at the point x 0 there are finite And , but f(x 0 +0)≠f(x 0 -0), then x 0 is called discontinuity point of the first kind , and is called function jump .

Example 2. Consider the function
The function can only be broken at the point x=2 (at other points it is continuous like any polynomial).
We'll find , . Since the one-sided limits are finite, but not equal to each other, then at the point x=2 the function has a discontinuity of the first kind. notice, that , therefore the function at this point is continuous on the right (Fig. 2).
2) Discontinuity points of the second kind are called points at which at least one of the one-sided limits is equal to ∞ or does not exist.

Example 3. The function y=2 1/ x is continuous for all values ​​of x except x=0. Let's find one-sided limits: , , therefore x=0 is a discontinuity point of the second kind (Fig. 3).
3) Point x=x 0 is called removable break point , if f(x 0 +0)=f(x 0 -0)≠f(x 0).
We will “eliminate” the gap in the sense that it is enough to change (redefine or redefine) the value of the function at this point by setting , and the function will become continuous at the point x 0 .
Example 4. It is known that , and this limit does not depend on the way x tends to zero. But the function at point x=0 is not defined. If we redefine the function by setting f(0)=1, then it turns out to be continuous at this point (at other points it is continuous as the quotient of the continuous functions sinx and x).
Example 5. Examine the continuity of a function .
Solution. The functions y=x 3 and y=2x are defined and continuous everywhere, including in the indicated intervals. Let's examine the junction point of the intervals x=0:
, , . We obtain that , which implies that at the point x=0 the function is continuous.
Definition. A function that is continuous on an interval except for a finite number of points of discontinuity of the first kind or removable discontinuity is called piecewise continuous on this interval.

Examples of discontinuous functions

Example 4. Is it possible to eliminate function gaps:
a) at point x=2;
b) at point x=2;
V) at point x=1?
Solution. Regarding example a) we can immediately say that the discontinuity f(x) at the point x=2 cannot be eliminated, since at this point there are infinite one-sided limits (see example 1).
b) The function g(x) although has finite one-sided limits at the point x=2

(,),

Example 5. Show that the Dirichlet function

Example 6. Find function breakpoints

For independent decision.
Examine functions for continuity and determine the type of discontinuity points:
1) ; Answer: x=-1 – point of removable discontinuity;
2) ; Answer: Discontinuity of the second kind at point x=8;
3) ; Answer: Discontinuity of the first kind at x=1;
4)
Answer: At the point x 1 =-5 there is a removable gap, at x 2 =1 there is a gap of the second kind and at the point x 3 =0 there is a gap of the first kind.
5) How should the number A be chosen so that the function

would be continuous at x=0?
Answer: A=2.
6) Is it possible to choose the number A so that the function

would be continuous at x=2?
Answer: no.

Lesson objectives:

Develop knowledge, skills and abilities to effectively use the generalized method of intervals, based on the property of continuous functions;

Formulate an algorithm of actions leading to equivalent transformations;

Teach how to independently apply it when solving inequalities;

Transfer knowledge, skills and abilities to new conditions.

Educational: systematization, consolidation, generalization of knowledge, skills and abilities.

Educational: nurturing the need for full-fledged consistent argumentation, accuracy, and independence.

Developmental: development of mathematical logic, formation of a mathematical style of thinking (clear dissection of the course of reasoning), cognitive interest.

1) Introduction, setting the goals and objectives of the lesson - 2 min.

2) Checking homework - 2 min.

(frontal work, self-control).

3) Mathematical justification of the stages of solving inequalities using the interval method - 4 minutes (prepared student answers).

4) Repetition of properties of inequalities – 2 min.

5) Preparation for mastering (studying) new educational material through repetition and updating of basic knowledge – 5 min. (front work, answers to questions, problem situations).

6) Generalized method of intervals for solving inequalities, initial comprehension – 13 min.

(collective solution of inequalities using the interval method: on the board and in notebooks).

7) Information about homework, instructions for completion – 1 min.

8) Consolidation of new knowledge – 15 min.

1) The need for a wider use of the interval method in school is dictated by the ideology of the entire process of teaching mathematics. The point is that the functional line (one of the main ones when studying the fundamentals of mathematics) receives powerful technological support. The interval method is based on such important characteristics of functional dependence as zeros of the function, intervals of its constant sign and monotonicity. Then the functional origin of equations and inequalities, as well as methods for solving them, becomes more clear. The categories of continuity of a function, the behavior of its graph in the vicinity of points of infinite discontinuity, theorems on roots, constancy of sign, extreme points and their types become more clear. And all this is organically linked into one functional whole.

On the other hand, the geometrization of the research objects used is also invaluable, i.e. visually and figuratively represent all the mathematical tools of functional dependence used.

The basic principles underlying the interval method:

• functional (generalized) approach;
• reliance on geometrization of functional properties;
• research visualization.

This leads to the following advantages of the method compared to others used in the same type of tasks: simplicity and speed of achieving the goal; visibility (and the ability to control or double-check); economy in computing resources and time; breadth of coverage of the entire situation, the formation and development of skills of generalized thinking and analysis, as well as the associated ability to draw logical conclusions.

2) Checking homework.(Slide No. 4)

3) A story about the interval method for solving inequalities. (Students' answers).

Mathematical justification for solving inequalities using the interval method.

1) Consider the inequalities: (x-2)(x-3)>0. (slide number 5)

You can solve it this way: The product (quotient) of two factors is positive if and only if both factors are of the same sign, i.e. inequality is equivalent to the combination of two systems: (slide No. 6)

From the first system we obtain x >3, from the second x< 2.

The solution is to combine the solutions of the two systems.

Answer:

Graphic method (slide number 7)

Another method is interval method(slide number 8).

His idea is as follows.

Let us mark on the number line the zeros (roots) of the polynomial (x-2)(x-3) standing

on the left side of the inequality, i.e. numbers 2 and 3.

When x >3 (to the right of the larger root), then (x-2)(x-3)>0, since each factor is positive.

If you move along the axis in a negative direction, then when passing through the point x=3, the factor (x-3) will change sign. In the product (x-2)(x-3) one negative factor will appear, resulting in (x-2)(x-3)<0. При переходе через следующий корень появится еще один отрицательный множитель и произведение (х-2)(х-3)>0.

Now it’s easy to write the solution to the inequality:

Conclusion: the product can change sign only when passing through the points x=2 and x=3

and, therefore, preserves the sign on each of the resulting intervals.

From this simple example, it is easy to understand the idea of ​​the interval method, but it does not show its significant advantages.

Let us consider the rationality of the interval method and its power in the following example (slide No. 9, 10,11, 12))

2) Solve the inequality (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)( x-10)>0.

To solve this inequality using a set of systems, one would have to consider a set of 512 systems with 10 inequalities in each system.

Let's use the interval method. Let us mark the zeros of the polynomial on the number line. On the interval x>10 the polynomial will be positive, since each factor is positive. When passing through each subsequent root, the polynomial will change sign, since an additional negative factor will appear in the product. Now it is easy to write down the solution to the inequality using alternating signs.

Advantages of the interval method.

• simplicity and speed of achieving the goal;
• visibility (and the ability to control or double-check);
• significant reduction in the amount of computational work and time;
• breadth of coverage of the entire situation;
• formation and development of skills of generalized thinking and analysis, as well as the associated ability to make logical conclusions.

Comment. It is very convenient to solve inequalities, the left side of which is factorized, since it is not difficult to find the zeros (roots).

Assignment: Solve inequality using the interval method (x+3) 3 (x-4) 2 (x-5)>0(Slide 13)

4) Repetition of properties of inequalities.

a) Question: What inequalities are called equivalent?

(Two inequalities are said to be equivalent if any solution to the first inequality is a solution to the second and, conversely, any solution to the second is a solution to the first).

Or: two inequalities are called equivalent if the sets of their solutions coincide.

Slide 14. Repetition of properties of inequalities.

Slide 15. Answer the question and explain.

Are inequalities equivalent?

1) 4x-5<0 и 4х<5

2) -2x+5>0 and 2x-5<0

3) -3x 2 +5x-7>0 and 3x 2 -5x+7<0

4) (x+1)>0 and (x 2 +5x+10)(x+1)>0

5) Oral frontal work to prepare for the assimilation (study) of new educational material through repetition and updating of basic knowledge.

Slide 16. Definition of a function continuous at a point.

Slide 17. Property of continuous functions.

Slide 18. Find intervals of continuity.

Slide 19. Find the mistake.

Slide 20. Solve the inequality orally,
using a chart.

Slide 21, 22. Replacing inequality with an equivalent condition.

Solve inequality

This inequality is equivalent to the condition f(x) < 0, counting

Therefore, we need to find all values ​​of x for which the condition f(x) is satisfied < 0.

6) Generalized method of intervals for solving inequalities, initial comprehension – 10 min. (collective solution of inequalities using the interval method: on the board and in notebooks).

Slide 23. Algorithm. Generalized method for solving inequalities.

Solving inequalities f(x)>0, f(x) > 0, f(x)<0, f(x)< 0 by interval method. (Scheme)

Slides 24 and 25. Solving inequalities using an algorithm. (Comments to all points of the algorithm).

Slide 26. Graphic illustration of the solution to this inequality.

Slide 27. Solve inequalities on the board and in notebooks .

Slide 28. Graphic illustration of the solution to this inequality.

Slide 29. Solve inequalities on the board and in notebooks

Slide 30. Graphic illustration of the solution to this inequality.

Slide 31, 32. Solve the inequality orally, using a picture

7) Information about homework.(Solve by interval method option No. 2)

8) Consolidation of new knowledge (independent work, option No. 1).

9) Summing up the lesson, self-control using ready-made solutions (slides 33, 34, 35), repeating the algorithm of the generalized interval method and its application.

10) Analysis of student learning and interest in the topic. This method is universal for solving any inequalities, including rational, modulus, irrational, exponential, logarithmic, since the method of intervals reduces the solution of inequalities to the solution of equations; finding the domain of definition and the value of a function at a point does not cause difficulties. But I had to give examples of inequalities where the use of this method is not justified, where it is more rational to use other methods for solving inequalities.

Presentation “Application of continuity in solving inequalities.” (35 slides)

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 weird? 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 seen:

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 chart.

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

1)

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.

Definition 4. A function is called continuous on a segment if it is continuous at every point of this segment (at point a it is continuous on the right, i.e., and at point b it is continuous on the left, i.e.).

All basic elementary functions are continuous in the domain of their definition.

Properties of functions continuous on an interval:

• 1) If a function is continuous on an interval, then it is bounded on this interval (Weierstrass's first theorem).
• 2) If a function is continuous on a segment, then on this segment it reaches its minimum value and maximum value (Weierstrass’s second theorem) (see Fig. 2).
• 3) If a function is continuous on a segment and takes on values ​​of different signs at its ends, then inside the segment there is at least one point such that (Bolzano-Cauchy theorem).

Function break points and their classification

function continuity point segment

The points at which the continuity condition is not satisfied are called break points of this function. If is a discontinuity point of a function, then at least one of the three conditions for the continuity of a function specified in Definitions 1, 2 is not satisfied, namely:

1) The function is defined in a neighborhood of a point, but not defined at the point itself. So the function considered in example 2 a) has a discontinuity at a point, since it is not defined at this point.

2) The function is defined at a point and its surroundings, there are one-sided limits and, but they are not equal to each other: . For example, the function from example 2 b) is defined at a point and its vicinity, but, since a.

3) The function is defined at the point and its surroundings, there are one-sided limits and they are equal to each other, but not equal to the value of the function at the point: . For example, a function. Here is the break point: at this point the function is defined, there are one-sided limits and, equal to each other, but, i.e. .

Function break points are classified as follows.

Definition 5. A point is called a discontinuity point of the first kind of function if at this point there are finite limits and, but they are not equal to each other: . The quantity is called the jump of the function at a point.

Definition 6. A point is called a point of removable discontinuity of a function if at this point there are finite limits and, they are equal to each other: , but the function itself is not defined at the point, or is defined, but.

Definition 7. A point is called a discontinuity point of the second kind of function if at this point at least one of the one-sided limits (or) does not exist or is equal to infinity.

Example 3. Find the breakpoints of the following functions and determine their type: a) b)

Solution. a) The function is defined and continuous on intervals, and, since on each of these intervals it is defined by continuous elementary functions. Consequently, the break points of a given function can only be those points at which the function changes its analytical task, i.e. points and Let's find the one-sided limits of the function at the point:

Since one-sided limits exist and are finite, but are not equal to each other, the point is a discontinuity point of the first kind. Function jump:

For the point we find.