Математика. Материалы курса

The topic of the lecture is continuity of a function at a point. The concept is extremely important, and it originated two hundred years ago. In 1817, the first definition of continuity in the language of ε-Δ was created by Bolzano. What we know about the continuity is that it is a continuous curve that can be built without taking our hands off.

How to translate this concept into the language of symbols? Let’s see. First, the function must be defined at the point x₀. The definition of the continuity in the language of ε-Δ can be written using neighborhood symbolism – this is the first definition, and, secondly, in the language of inequalities.

Let’s look at the definitions (see the video). They are very similar to the definition of the limit of a function at the point x₀, except that in the last implication f(x) belongs to the ε-neighborhood of f(x₀), and the neighborhood of x₀ is not punctured. How do we understand this definition? How do we use it? This is certainly not an easy task to do.

We will consider four definitions of continuity. This is the first one (see the video). It was in this form that it was formed in the first half of the 19th century in the language of sequences, just like the definition of a limit. A function is called a continuous at the point x₀ if whatever the sequence x_n made up of points in the domain of definition, if it converges to the point х₀, the corresponding sequence of function values converges to the number f(x₀).

Those two definitions are common.  х₀ was not required to be  the limit point of the definition domain. In fact, it could be an isolated point. Definitions 3, 4 are applied only if х₀ is the limit point of the set D(f). This definition is called a definition in the language of limits. As a matter of fact, we have it in definition 1 and definition 2.

A function is said to be continuous at x₀ if the limit of the function at this point is equal to the value of the function. That’s a very important definition which we will use in the future when we talk about points of discontinuity.

Let’s analyze it. There are actually three conditions here. Existence of the left part. The limit of the function at the point x₀ must exist and be a number. The second condition is: the function must be defined at the point x₀, there is a right-hand side of f (x₀). And the third condition is: these two numbers (the left and right parts) must be equal.

We will need definition 4 in the next section, we will study it soon.

Note the point x₀ and f(x₀) on the graph. Then we take the following steps. If we give the argument a change, we say that the increment Δ x is not zero, we get a new point x₀+Δx on the x-axis. Let’s see what happens on the y-axis. The value of the function will also change by the value that f(x₀) became f(x₀+Δx₀). The value of change (see the video) is called the increment of the function f(x₀). So, the increment of the argument on the x-axis, the increment of the function on the y-axis. We get two increments.

Let us recall the definition in the language of limits. The function is continuous at the point x₀ if the limit at point 0 is equal to the number f(x₀). Let us reconstruct the proofs and come to definition 4. It is quite obvious that in this case the limit of the difference between f(x) and f(x₀) is zero. The new notation Δx is the difference x minus x₀. It is clear that Δx tends to 0 if x tends to x₀. We express x from the replacement.

What do we get in this case? The previous entry gets a new look (see the video), and it is easy to see that the limit is calculated from the increment of the function at the point x₀. We got a new definition. A function is said to be continuous at the point x₀ if the limit of the increment of the function at the point x₀ is zero when Δx tends to zero. This definition is frequently read as follows: an infinitesimal increment of the function corresponds to an infinitesimal increment of the argument.

Let’s see how this definition works for a simple function f(x)=x². Let x₀ be an arbitrary point of the numeric line. Let’s give it an increment of Δx. We get a new point x₀+Δx, make an increment of the function Δf(x₀) by the formula, calculate by substituting the value x₀+Δх for the function expression instead of x, open the brackets and get it (see the video). Note, a function is continuous if the limit of the function increment is zero when Δx tends to zero, which is what we get. So, Δx is a variable that tends to zero, x₀ is a number, and the answer is 0. The conclusion is that this function is continuous at any point on the numeric line.

We are talking about continuity of functions in general or on a set, what do we mean by this? A function is continuous on a set if it is continuous at any point in that set. If the set is not mentioned, we are talking about continuity over the entire domain of definition. The graph of a continuous function, respectively, is a continuous curve. The previous example tells us that the function y=x² is continuous. We remember that the graph of a quadratic function is a parabola, so a parabola is a continuous curve.

We are not going to prove the continuity of the other elementary functions. First, all the basic elementary functions are continuous, the proof is sometimes difficult, but it is actually carried out and proved. What is an elementary function? This is a function that is obtained using basic elementary functions when we apply a finite number of arithmetic operations of addition and multiplication, subtraction, division, and composition of functions to them. So it is proved that all elementary functions are continuous, that is, continuous on their domains of definitions. We will use this crucial fact.

What is a point of discontinuity? This is the point x₀ where the definition of continuity is broken. Let’s use the third definition that we had in the language of limits, that is, the limit of a function at a point x₀ is not equal to the number f(x₀).  We mean that x₀ is the limit point of the definition domain on both the left and right sides.

Let’s consider the classification of points of discontinuity. Let x₀ be the point of discontinuity, that is, the specified equality does not hold. What are the reasons for this? One-sided limits at the point x₀ are finite, they can be equal or unequal, it doesn’t matter, this is a point of discontinuity. In this case, x₀ is called a point of discontinuity of the first kind. If we construct a negation, which means that the one-sided limits at x₀ are not finite. We get a point of discontinuity of the second kind. In what case? At least one of the one-sided limits is infinite or does not exist, in this case x₀ is called a point of discontinuity of the second kind.

We consider the classification of points of discontinuity of the first kind. Let’s deal with one-sided limits. What can it be? One-sided limits can be equal numbers, or they can be unequal numbers. If these two one-sided limits are equal, the gap is called removable, and if they are not equal, the gap is called irremovable. We say that the function at x₀ has a jump. We have already mentioned that a function is continuous on a set if it is continuous at every point of that set.

Theorems on functions that are continuous on a segment are of great importance. The authorship of these theorems belongs to mathematicians (see the video). We have already mentioned, Bolzano was the first to talk about the definition of continuity. There are two more names of mathematicians – Cauchy and Weierstrass. So, it is the first Bolzano-Cauchy theorem. It is briefly called the zeros theorem of a continuous function on a segment. If a function is continuous on a segment and takes different sign values at the ends of this segment, plus at one end and minus at the other, there is a point belonging to this segment where the function value is zero. This is the first theorem from the theorems about functions that are continuous on a segment.

The second theorem is also Bolzano-Cauchy theorem on intermediate values of continuous functions on a segment. If the function is continuous on the segment [a, b], any number r enclosed between the values of the function at the ends of f(a) and f(b) is also the value of the function at some point in this segment. That is, there is a point c that belongs to the segment where the value is equal to this number r.

The third and fourth theorems belong to Karl Weierstrass. The first of them is the boundedness theorem of a continuous function on a given segment. If a function is continuous on a segment, it is bounded on it.

And another Weierstrass theorem is if the function is continuous on a segment, it reaches its largest and smallest values on it. That is, there is a point where the value of the function at this point is the largest of all the values accepted by the function on the segment. Similarly, there is a point belonging to a segment where the value is the smallest of all accepted functions on the segment.