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How to investigate a function for continuity?




The study of the function of continuity at a point is carried out according to an already rolled-up routine, which consists of checking three continuity conditions:

Example 1

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

Solution :

1) A single point falls under the scope in which the function is not defined.

2) We calculate the one-sided limits:

One-sided limits are finite and equal.

So at the point the function suffers a removable gap.

What does the graph of this function look like?

I want to simplify , and it seems to be an ordinary parabola. BUT the original function is not defined at the point therefore the following disclaimer is mandatory:

Let's execute the drawing:

Answer : the function is continuous on the whole number line except for the point in which she suffers a removable break.

The function can be redefined in a good or not very way, but by condition this is not required.

You say a far-fetched example? Not at all. Dozens of times met in practice. Almost all the tasks of the site come from real independent and control work.

We will deal with our favorite modules:

Example 2

Explore function on continuity. Determine the nature of the discontinuities of the function, if they exist. Run the drawing.

Solution : for some reason, students are afraid and do not like functions with the module, although there is nothing complicated in them. We already touched upon 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 sign in a piecewise way:

But the fractions of both pieces have to be reduced by . The reduction, as in the previous example, will not go without consequences. Source function not defined at point , since the denominator vanishes. Therefore, the system should additionally indicate the condition , and the first inequality make strict:

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

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

If in doubt, take a few "X" values, substitute them in a function (not forgetting that the module destroys the possible minus sign) and check the schedule.


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We study the continuity function analytically:

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

2) We establish the nature of the gap, for this we calculate the one-sided limits:

One-sided limits are finite and different, which means that the function is discontinuous of the first kind with a jump at a point . Note that it does not matter if the function is defined at the break point or not.

Now it remains 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 whole number line except for the point in which she suffers a break of the first kind with a jump.

Sometimes it is required to additionally indicate a jump in the gap. It is calculated elementarily - from the right limit you need to subtract the left limit: , that is, at the break point, our function jumped 2 units down (as indicated by the minus sign).

Example 3

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

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

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

Example 4

Investigate the function for continuity and plot the function

.

Solution : it is obvious that all three parts of the function are continuous at the corresponding intervals, so it remains to check only two points of “junction” between the pieces. First, we will draw the draft on the draft, I commented on the construction technique in sufficient detail in the first part of the article. The only thing you need to carefully follow our special points: due to inequality value belongs to direct (green dot), and due to inequality value belongs to the parabola (red dot):

Well, in principle, everything is clear =) It remains to draw up a solution. For each of the two “butt” points, we routinely check 3 continuity conditions:



I) We investigate the continuity of the point

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

2) Find the one-sided limits:


One-sided limits are finite and different, which means that the function suffers a break of the first kind with a jump at a point .

We calculate the gap jump as the difference between the right and left limits:
, that is, the graph pulled one unit up.

II) We investigate the continuity of the point

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

2) Find the one-sided limits:

- unilateral limits are finite and equal, which means that there is a common limit.

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

So the function continuous at point by the definition of the continuity of a function at a point.

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

Answer : the function is continuous on the whole number line, except for the point in which she suffers a break of the first kind with a jump.

Done.

Example 5

Investigate a function for continuity and plot it .

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

It may seem that at one point the function must be continuous, and at the other, there must be a gap. In practice, this is far from always the case. Try not to neglect the remaining examples - there will be some interesting and important chips:

Example 6

Given function . Investigate the function of continuity at points . Build a schedule.

Solution : and again immediately execute the draft drawing:

The peculiarity of this graph is that when piecewise function is given by the equation of the abscissa axis . Here, this section is drawn in green, and in a notebook it is usually boldly highlighted with a simple pencil. And, of course, do not forget about our sheep: the meaning refers to the tangent branch (red dot), and the value belongs to direct .

Everything is clear from the drawing - the function is continuous on the whole numerical line, it remains to draw up a solution that is brought to full automatism literally after 3-4 such examples:

I) We investigate the continuity of the point

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

2) We calculate the one-sided limits:

, then a common limit exists.

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

We go further:

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

So the function continuous at point by the definition of the continuity of a function at a point.

II) We investigate the continuity of the point

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

2) Find the one-sided limits:

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

- a common limit exists.

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

So the function continuous at point by the definition of the continuity of a function at a point.

As usual, after research, we transfer our drawing to a clean copy.

Answer : the function is continuous at points .

Please note that in the condition we were not asked anything about the study of the entire function for continuity, and it is considered to be a good mathematical tone to formulate an accurate and clear answer to the question posed. By the way, if the condition does not require the construction of a schedule, then you have every right to not build it (however, then the teacher can make it happen).

A small mathematical “tongue twister” for an independent solution:

Example 7

Given function .

Investigate the function of continuity at points . Classify break points, if any. Run the drawing.

Try to correctly "pronounce" all the "words" =) And draw a more accurate schedule, accuracy, it will not be superfluous everywhere ;-)

As you remember, I recommended that you immediately draw a draft on a draft, but from time to time you come across examples where you can’t immediately figure out how the chart looks. Therefore, in some cases, it is advantageous to first find one-sided limits and only then, on the basis of research, depict branches. In the two final examples, in addition, we will master the technique of calculating some one-sided limits:

Example 8

Explore Continuity Function and build her schematic diagram.

Solution : bad points are obvious: (sets the denominator of the indicator to zero) and (nullifies the denominator of the whole fraction). It is not clear what the graph of this function looks like, which means that it is better to conduct a study first:

I) We investigate the continuity of the point

1) The function is not defined at this point.

2) Find the one-sided limits:

Pay attention to the typical method of calculating the one-way limit : we substitute the function “x” in the function . There is no crime in the denominator: “addition” “minus zero” does not play a role, and it turns out “four”. But in the numerator there is a small thriller: first in the denominator of the indicator kill –1 and 1, resulting in . The unit divided by an infinitesimal negative number is equal to "minus infinity", therefore: . And finally, the "deuce" in an infinitely large negative degree is equal to zero: . Or, if more in detail: .

We calculate the right-hand limit:

And here - instead of “X” we substitute . In the denominator is “additive” again does not matter: . In the numerator, actions similar to the previous limit are carried out: we destroy the opposite numbers and divide the unit by an infinitely small positive number :

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

II) We investigate the continuity of the point

1) The function is not defined at this point.

2) We calculate the left-side limit:

The method is the same: we substitute “x” into the function . There is nothing interesting in the numerator - we get a finite positive number . And in the denominator, we open the brackets, remove the “triples”, and the “additive” plays a decisive role .

As a result, a finite positive number divided by an infinitesimal positive number gives "plus infinity": .

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

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

Thus, we have two break points, and, obviously, three branches of the chart. For each branch, it is advisable to carry out pointwise construction, i.e. take a few X values ​​and substitute them in . чертежа, и такое послабление естественно для ручной работы. Note that, by condition, a schematic drawing is allowed, and such relaxation is natural for manual work. I build graphs using the program, so I have no such difficulties, here is a fairly accurate picture:

Direct are the vertical asymptotes for the graph of this function.

Answer : the function is continuous on the whole number line except points in which she suffers breaks of the 2nd kind.

A simpler function for an independent solution:

Example 9

Explore Continuity Function and perform a schematic drawing.

An approximate sample of the solution at the end, which crept unnoticed.

See you soon!

Decisions and answers:

Example 3: Solution : transform the function: . Given the module disclosure rule and the fact that , we rewrite the function in piece form:

We study the function for continuity.

1) The function is not defined at the point .

2) We calculate the one-sided limits:


One-sided limits are finite and different, which means that the function is discontinuous of the first kind with a jump at a point . Let's execute the drawing:

Answer : the function is continuous on the whole number line except for the point in which she suffers a break of the first kind with a jump. Gap jump: (two units up).

Example 5: Solution : each of the three parts of the function is continuous on its interval.
I) We investigate the continuity of the point
one) - the function is defined at a given point.

2) We calculate the one-sided limits:


, then a common limit exists.
3) - the limit of the function at a point is equal to the value of this function at a given point.
So the function continuous at point by the definition of the continuity of a function at a point.
II) We investigate the continuity of the point

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

2) Find the one-sided limits:


One-sided limits are finite and different, which means that the function suffers a break of the first kind with a jump at a point .
Gap jump: (five units down).
The drawing can be found in the first part of the article.
Answer : the function is continuous on the whole number line, except for the point in which she suffers a break of the first kind with a jump.

Example 7: Solution :

I) We investigate the continuity of the point

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

2) Find the one-sided limits:


The left-hand limit is infinite, which means that the function suffers a discontinuity of the second kind at the point .
II) We investigate the continuity of the point

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

2) Find the one-sided limits:


One-sided limits are finite and different, which means that the function suffers a break of the first kind with a jump at a point .
Let's execute the drawing:

Answer : At a point the function suffers a gap of the 2nd kind, at the point the function breaks the 1st kind with a jump.

Example 9: Solution : examine the continuity point :

1) The function is not defined at this point.

2) We calculate the one-sided limits:


The left-hand limit is infinite, which means that the function suffers a discontinuity of the second kind at the point .
Let's execute the drawing:

Answer : the function is continuous on the whole number line except for the point in which she suffers a gap of the 2nd kind.

Author: Emelin Alexander

Higher mathematics for external students and not only >>>

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How to find the scope of a function?

Solution Examples

If somewhere there isn’t something, then somewhere there is something

We continue to study the section "Functions and Graphics", and the next station of our journey is the Function Definition Area . An active discussion of this concept began in the very first lesson on function graphs , where I examined elementary functions, and, in particular, their domain of definition. Therefore, I recommend teapots to start with the basics of the topic, as I will not again dwell on some basic points.

It is assumed that the reader knows the domain of definition of the main functions: linear, quadratic, cubic function, polynomials, exponent, logarithm, sine, cosine. They are defined on . For tangents, arcsines, so be it, I forgive =) Rarer graphics are remembered far from immediately.

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

Let's start with the express cut in the topic. Briefly about the main thing: we are talking about the function of one variable . Its domain of definition is the set of “X” meanings for which there are meanings of “gamers”. Consider a conditional example:

The scope of this function is the union of the gaps:
(for those who have forgotten: - association 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 meaning of "game".

Roughly speaking, where is the domain of definition - there is a graph of the function. And here is the half-interval and the point "ce" is not included in the definition area, so there is no graph there.

Yes, by the way, if something 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 .

How to find the scope of a function? Many people remember the children's room: “stone, scissors, paper”, and in this case it can be safely rephrased: “root, fraction and logarithm”. Thus, if you encounter a fraction, a root or a logarithm on your life path, then you should be very, very alert at once! Tangent, cotangent, arcsine, arccosine are much less common, and we will also talk about them. But first, sketches from the life of ants:





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