Continuity of elementary functions

The continuity theorems for functions follow directly from the corresponding limit theorems.

Theorem. The sum, product and quotient of two continuous functions is a continuous function (for the quotient, except for those values ​​of the argument in which the divisor is zero).

Theorem. Let the functions u= φ (x) is continuous at the point X 0 , and the function y = f(u) is continuous at the point u 0 = φ (X 0). Then the complex function f(φ (x)) consisting of continuous functions is continuous at the point x 0 .

Theorem. If the function at = f(X) is continuous and strictly monotone on [ a; b] axis Oh, then the inverse function at = φ (X) is also continuous and monotone on the corresponding segment [ c;d] axis OU(no proof).

Functions continuous on an interval have a number of important properties. We formulate them in the form of theorems without giving proofs.

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

The function shown in Figure 5 at = f(x) is continuous on the segment [ a; b], takes its maximum value M at the point x 1 , and the least m- at the point X 2. For anyone X [a; b] m ≤ f(x) ≤ M.

Consequence. If a function is continuous on an interval, then it is bounded on this interval.

Theorem (Bolzano - Cauchy). If the function at= f(x) is continuous on the segment [ a; b] and takes unequal values ​​at its ends f(a) = A and f(b) = =AT, then on this segment it also takes on all intermediate values ​​between BUT and AT.

Geometrically, the theorem is obvious (see Fig. 6).

For any number FROM concluded between BUT and AT, there is a point With inside this segment such that f(With) = FROM. Straight at = FROM intersects the graph of the function at least at one point.

Consequence. If the function at = f(x) is continuous on the segment [ a; b] and takes on values ​​of different signs at its ends, then inside the segment [ a; b] there is at least one point With, in which this function f(x) vanishes: f(With) = 0.

The geometric meaning of the theorem: if the graph of a continuous function passes from one side of the axis Oh to another, then it crosses the axis Ox(See Fig. 7).

Rice. 7.

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 their domain of definition.

Properties of functions continuous on a segment:

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

Function breakpoints and their classification

function continuity point segment

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

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

2) The function is defined at a point and its neighborhood, 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 neighborhood, 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, 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 breakpoints are classified as follows.

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

Definition 6 . A point is called a point of a 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 a 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 break points of the following functions and determine their type: a) b)

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

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

For a point we find.

PROPERTIES OF FUNCTIONS CONTINUOUS ON A INTERVAL

Let us consider some properties of functions continuous on an interval. We present these properties without proof.

Function y = f(x) called continuous on the segment [a, b], if it is continuous at all internal points of this segment, and at its ends, i.e. at points a and b, is continuous on the right and left, respectively.

Theorem 1. A function continuous on the segment [ a, b], at least at one point of this segment takes the largest value and at least at one point - the smallest.

The theorem states that if the function y = f(x) continuous on the interval [ a, b], then there is at least one point x 1 Î [ a, b] such that the value of the function f(x) at this point will be the largest of all its values ​​on this segment: f(x1) ≥ f(x). Similarly, there is such a point x2, in which the value of the function will be the smallest of all values ​​on the segment: f(x 1) ≤ f(x).

It is clear that there can be several such points, for example, the figure shows that the function f(x) takes the smallest value at two points x2 and x 2 ".

Comment. The statement of the theorem can become false if we consider the value of the function on the interval ( a, b). Indeed, if we consider the function y=x on (0, 2), then it is continuous on this interval, but does not reach its maximum or minimum values ​​in it: it reaches these values ​​at the ends of the interval, but the ends do not belong to our region.

Also, the theorem ceases to be true for discontinuous functions. Give an example.

Consequence. If the function f(x) continuous on [ a, b], then it is bounded on this segment.

Theorem 2. Let the function y = f(x) continuous on the segment [ a, b] and takes on values ​​of different signs at the ends of this segment, then there is at least one point inside the segment x=C, where the function vanishes: f(C)= 0, where a< C< b

This theorem has a simple geometric meaning: if the points of the graph of a continuous function y = f(x), corresponding to the ends of the segment [ a, b] lie on opposite sides of the axis Ox, then this graph at least at one point of the segment intersects the axis Ox. Discontinuous functions may not have this property.

This theorem admits the following generalization.

Theorem 3 (theorem on intermediate values). Let the function y = f(x) continuous on the segment [ a, b] and f(a) = A, f(b) = B. Then for any number C between A and B, there is such a point inside this segment CÎ [ a, b], what f(c) = C.

This theorem is geometrically obvious. Consider the graph of the function y = f(x). Let f(a) = A, f(b) = B. Then any line y=C, where C- any number between A and B, intersects the graph of the function at least at one point. The abscissa of the intersection point will be that value x=C, at which f(c) = C.

Thus, a continuous function, passing from one of its values ​​to another, necessarily passes through all intermediate values. In particular:

Consequence. If the function y = f(x) is continuous on some interval and takes on the largest and smallest values, then on this interval it takes, at least once, any value between its smallest and largest values.

DERIVATIVE AND ITS APPLICATIONS. DERIVATIVE DEFINITION

Let's have some function y=f(x), defined on some interval. For each argument value x from this interval the function y=f(x) has a certain meaning.

Consider two argument values: initial x 0 and new x.

Difference x–x 0 is called increment of argument x at the point x 0 and denoted Δx. In this way, ∆x = x – x 0 (argument increment can be either positive or negative). From this equality it follows that x=x 0 +Δx, i.e. the initial value of the variable has received some increment. Then, if at the point x 0 function value was f(x 0 ), then at the new point x the function will take the value f(x) = f(x 0 +∆x).

Difference y-y 0 = f(x) – f(x 0 ) called function increment y = f(x) at the point x 0 and is denoted by the symbol Δy. In this way,

Δy = f(x) – f(x 0 ) = f(x 0 +Δx) - f(x 0 ) .

(1)

Usually the initial value of the argument x 0 is considered fixed and the new value x- variable. Then y 0 = f(x 0 ) turns out to be constant and y = f(x)- variable. increments Δy and Δx will also be variables and formula (1) shows that Dy is a function of the variable Δx.

Compose the ratio of the increment of the function to the increment of the argument

Let us find the limit of this relation at Δx→0. If this limit exists, then it is called the derivative of this function. f(x) at the point x 0 and denote f "(x 0). So,

derivative this function y = f(x) at the point x 0 is called the limit of the increment ratio of the function Δ y to the increment of the argument Δ x when the latter arbitrarily tends to zero.

Note that for the same function the derivative at different points x can take on different values, i.e. the derivative can be thought of as a function of the argument x. This function is denoted f "(x)

The derivative is denoted by the symbols f "(x),y", . The specific value of the derivative at x = a denoted f "(a) or y "| x=a.

The operation of finding the derivative of a function f(x) is called the differentiation of this function.

To directly find the derivative by definition, you can apply the following rule of thumb:

Examples.

MECHANICAL MEANING OF THE DERIVATIVE

It is known from physics that the law of uniform motion has the form s = v t, where s- path traveled up to the point in time t, v is the speed of uniform motion.

However, since most of the movements occurring in nature are uneven, then in the general case, the speed, and, consequently, the distance s will depend on time t, i.e. will be a function of time.

So, let the material point move in a straight line in one direction according to the law s=s(t).

Note a moment in time t 0 . By this point, the point has passed the path s=s(t 0 ). Let's determine the speed v material point at time t 0 .

To do this, consider some other moment in time t 0 + Δ t. It corresponds to the distance traveled s =s(t 0 + Δ t). Then for the time interval Δ t the point has traveled the path Δs =s(t 0 + Δ t)–s(t).

Let's consider the relationship. It is called the average speed in the time interval Δ t. The average speed cannot accurately characterize the speed of movement of a point at the moment t 0 (because the movement is uneven). In order to more accurately express this true speed using the average speed, you need to take a smaller time interval Δ t.

So, the speed of movement at a given time t 0 (instantaneous speed) is the limit of the average speed in the interval from t 0 to t 0 +Δ t when Δ t→0:

,

those. speed of uneven movement is the derivative of the distance traveled with respect to time.

GEOMETRIC MEANING OF THE DERIVATIVE

Let us first introduce the definition of a tangent to a curve at a given point.

Let we have a curve and a fixed point on it M 0(see figure). Consider another point M this curve and draw a secant M 0 M. If point M starts to move along the curve, and the point M 0 remains stationary, the secant changes its position. If, with unlimited approximation of the point M curve to point M 0 on any side, the secant tends to take the position of a certain straight line M 0 T, then the straight line M 0 T is called the tangent to the curve at the given point M 0.

That., tangent to the curve at a given point M 0 called the limit position of the secant M 0 M when the point M tends along the curve to a point M 0.

Consider now the continuous function y=f(x) and the curve corresponding to this function. For some value X 0 function takes a value y0=f(x0). These values x 0 and y 0 on the curve corresponds to a point M 0 (x 0; y 0). Let's give an argument x0 increment Δ X. The new value of the argument corresponds to the incremented value of the function y 0 +Δ y=f(x 0 –Δ x). We get a point M(x 0+Δ x; y 0+Δ y). Let's draw a secant M 0 M and denote by φ the angle formed by the secant with the positive direction of the axis Ox. Let's make a relation and note that .

If now Δ x→0, then, due to the continuity of the function Δ at→0, and therefore the point M, moving along the curve, indefinitely approaches the point M 0. Then the secant M 0 M will tend to take the position of a tangent to the curve at the point M 0, and the angle φ→α at Δ x→0, where α denotes the angle between the tangent and the positive direction of the axis Ox. Since the function tg φ continuously depends on φ at φ≠π/2, then at φ→α tg φ → tg α and, therefore, the slope of the tangent will be:

those. f"(x)= tgα .

Thus, geometrically y "(x 0) represents the slope of the tangent to the graph of this function at the point x0, i.e. for a given value of the argument x, the derivative is equal to the tangent of the angle formed by the tangent to the graph of the function f(x) at the corresponding point M 0 (x; y) with positive axis direction Ox.

Example. Find the slope of the tangent to the curve y = x 2 at point M(-1; 1).

We have already seen that ( x 2)" = 2X. But the slope of the tangent to the curve is tg α = y"| x=-1 = - 2.

DIFFERENTIABILITY OF FUNCTIONS. CONTINUITY OF A DIFFERENTIABLE FUNCTION

Function y=f(x) called differentiable at some point x 0 if it has a certain derivative at this point, i.e. if the limit of the relation exists and is finite.

If a function is differentiable at every point of some segment [ a; b] or interval ( a; b), then they say that it differentiable on the segment [ a; b] or, respectively, in the interval ( a; b).

The following theorem is valid, which establishes a connection between differentiable and continuous functions.

Theorem. If the function y=f(x) differentiable at some point x0, then it is continuous at this point.

Thus, the differentiability of a function implies its continuity.

Proof. If a , then

,

where α is an infinitesimal value, i.e. quantity tending to zero at Δ x→0. But then

Δ y=f "(x0) Δ x+αΔ x=> Δ y→0 at Δ x→0, i.e. f(x) – f(x0)→0 at x→x 0 , which means that the function f(x) continuous at point x 0 . Q.E.D.

Thus, at discontinuity points, the function cannot have a derivative. The converse statement is not true: there are continuous functions that are not differentiable at some points (that is, they do not have a derivative at these points).

Consider the points in the figure a, b, c.

At the point a at Δ x→0 the relation has no limit (because the one-sided limits are different for Δ x→0–0 and Δ x→0+0). At the point A the graph has no defined tangent, but there are two different one-sided tangents with slopes to 1 and to 2. This type of point is called a corner point.

At the point b at Δ x→0 the ratio is of constant sign infinitely large value . The function has an infinite derivative. At this point, the graph has a vertical tangent. Point type - "inflection point" with a vertical tangent.

At the point c one-sided derivatives are infinitely large quantities of different signs. At this point, the graph has two merged vertical tangents. Type - "cusp" with a vertical tangent - a special case of a corner point.

From a practical point of view, the most interesting is the use of the derivative to find the largest and smallest value of a function. What is it connected with? Maximizing profits, minimizing costs, determining the optimal load of equipment... In other words, in many areas of life, one has to solve the problem of optimizing some parameters. And this is the problem of finding the largest and smallest values ​​of the function.

It should be noted that the largest and smallest value of a function is usually sought on some interval X , which is either the entire domain of the function or part of the domain. The interval X itself can be a line segment, an open interval , an infinite interval .

In this article, we will talk about finding the largest and smallest values ​​of an explicitly given function of one variable y=f(x) .

Page navigation.

The largest and smallest value of a function - definitions, illustrations.

Let us briefly dwell on the main definitions.

The largest value of the function , which for any the inequality is true.

The smallest value of the function y=f(x) on the interval X is called such a value , which for any the inequality is true.

These definitions are intuitive: the largest (smallest) value of a function is the largest (smallest) value accepted on the interval under consideration with the abscissa.

Stationary points are the values ​​of the argument at which the derivative of the function vanishes.

Why do we need stationary points when finding the largest and smallest values? The answer to this question is given by Fermat's theorem. It follows from this theorem that if a differentiable function has an extremum (local minimum or local maximum) at some point, then this point is stationary. Thus, the function often takes its maximum (smallest) value on the interval X at one of the stationary points from this interval.

Also, a function can often take on the largest and smallest values ​​at points where the first derivative of this function does not exist, and the function itself is defined.

Let's immediately answer one of the most common questions on this topic: "Is it always possible to determine the largest (smallest) value of a function"? No not always. Sometimes the boundaries of the interval X coincide with the boundaries of the domain of the function, or the interval X is infinite. And some functions at infinity and on the boundaries of the domain of definition can take both infinitely large and infinitely small values. In these cases, nothing can be said about the largest and smallest value of the function.

For clarity, we give a graphic illustration. Look at the pictures - and much will become clear.

On the segment

In the first figure, the function takes the largest (max y ) and smallest (min y ) values ​​at stationary points inside the segment [-6;6] .

Consider the case shown in the second figure. Change the segment to . In this example, the smallest value of the function is achieved at a stationary point, and the largest - at a point with an abscissa corresponding to the right boundary of the interval.

In figure No. 3, the boundary points of the segment [-3; 2] are the abscissas of the points corresponding to the largest and smallest value of the function.

In the open range

In the fourth figure, the function takes the largest (max y ) and smallest (min y ) values ​​at stationary points within the open interval (-6;6) .

On the interval , no conclusions can be drawn about the largest value.

At infinity

In the example shown in the seventh figure, the function takes the largest value (max y ) at a stationary point with x=1 abscissa, and the smallest value (min y ) is reached at the right boundary of the interval. At minus infinity, the values ​​of the function asymptotically approach y=3 .

On the interval, the function does not reach either the smallest or the largest value. As x=2 tends to the right, the function values ​​tend to minus infinity (the straight line x=2 is a vertical asymptote), and as the abscissa tends to plus infinity, the function values ​​asymptotically approach y=3 . A graphic illustration of this example is shown in Figure 8.

Algorithm for finding the largest and smallest values ​​of a continuous function on the segment .

We write an algorithm that allows us to find the largest and smallest value of a function on a segment.

1. We find the domain of the function and check if it contains the entire segment .
2. We find all points at which the first derivative does not exist and which are contained in the segment (usually such points occur in functions with an argument under the module sign and in power functions with a fractional-rational exponent). If there are no such points, then go to the next point.
3. We determine all stationary points that fall into the segment. To do this, we equate it to zero, solve the resulting equation and choose the appropriate roots. If there are no stationary points or none of them fall into the segment, then go to the next step.
4. We calculate the values ​​of the function at the selected stationary points (if any), at points where the first derivative does not exist (if any), and also at x=a and x=b .
5. From the obtained values ​​of the function, we select the largest and smallest - they will be the desired maximum and smallest values ​​of the function, respectively.

Let's analyze the algorithm when solving an example for finding the largest and smallest values ​​of a function on a segment.

Example.

Find the largest and smallest value of a function

• on the segment;
• on the interval [-4;-1] .

Solution.

The domain of the function is the entire set of real numbers, except for zero, that is, . Both segments fall within the domain of definition.

We find the derivative of the function with respect to:

Obviously, the derivative of the function exists at all points of the segments and [-4;-1] .

Stationary points are determined from the equation . The only real root is x=2 . This stationary point falls into the first segment.

For the first case, we calculate the values ​​of the function at the ends of the segment and at a stationary point, that is, for x=1 , x=2 and x=4 :

Therefore, the largest value of the function is reached at x=1 , and the smallest value – at x=2 .

For the second case, we calculate the values ​​of the function only at the ends of the segment [-4;-1] (since it does not contain a single stationary point):

Solution.

Let's start with the scope of the function. The square trinomial in the denominator of a fraction must not vanish:

It is easy to check that all intervals from the condition of the problem belong to the domain of the function.

Let's differentiate the function:

Obviously, the derivative exists on the entire domain of the function.

Let's find stationary points. The derivative vanishes at . This stationary point falls within the intervals (-3;1] and (-3;2) .

And now you can compare the results obtained at each point with the graph of the function. The blue dotted lines indicate the asymptotes.

This can end with finding the largest and smallest value of the function. The algorithms discussed in this article allow you to get results with a minimum of actions. However, it can be useful to first determine the intervals of increase and decrease of the function and only after that draw conclusions about the largest and smallest value of the function on any interval. This gives a clearer picture and a rigorous justification of the results.

Continuity of a function on a segment.

Along with the continuity of a function at a point, one considers its continuity on different intervals.

A function f (x) is called continuous on an interval (a, b) if it is continuous at every point of this interval.

A function f(x) is called continuous on the interval [a, b] if it is continuous on the interval (a, b), continuous on the right at point a, and continuous on the left at point b.

The function is called continuous on the segmentif it is continuous in the interval, continuous on the right at the point, that is and continuous on the left at the point, that is .

Comment. A function that is continuous on the segment [ a , b ] can be discontinuous at points a and b (Fig. 1)

The set of functions that are continuous on the segment [a, b] is denoted by the symbol C[a, b].

Basic theorems on functions continuous on an interval.

Theorem 1(on the boundedness of a continuous function). If the function f (x) is continuous on the segment [a, b], then it is bounded on this segment, i.e. there is a number C > 0 such that " x 0 [ a , b ] the inequality | f (x)| ≤ C .

Theorem 2(Weierstrass). If the function f (x) is continuous on the segment [a, b], then it reaches its maximum value M and its minimum value m on this interval, i.e. there are points α , β О [ a , b ] such that m = f (α) ≤ f (x) ≤ f (β) = M for all x О [ a , b ] (Fig. 2).

The largest value of M is denoted by the symbol max x About [a, b] f (x), and the smallest value of m is symbol min x About [a, b] f(x).
Theorem 3(on the existence of zero). If the function f (x) is continuous on the segment [ a , b ] and takes non-zero values ​​of different signs at the ends of the segment, then on the interval (a , b) there is at least one point ξ at which f (ξ) = 0.
The geometric meaning of the theorem is that the graph of a function that satisfies the conditions of the theorem will necessarily intersect the axis OX(Fig. 3).

Comment. This theorem is the basis for the method of approximate solution of the equation

f(x) = 0,

(1)

called the bisection (dichotomy) method, or the bisection method.

Theorem 4(Bolzano-Cauchy). If the function f (x) is continuous on the interval [a, b], then it takes on (a, b) all intermediate values ​​between f (a) and f (b).
Existence of a continuous inverse function
Let the function y = f (x) be defined, strictly monotonic and continuous on the segment [a, b]. Then on the interval [ α , β ] (α = f (a), β = f (b)) there exists an inverse function x = g (y), which is also strictly monotonic and continuous on the interval (α , β).