## Continuity

We have seen that any polynomial function P(x) satisfies:

for all real numbers a. This property is known as continuity.

Definition. Let f(x) be a function defined on an interval around a. We say that f(x) is continuous at a iff

Otherwise, we say that f(x) is discontinuous at a.

Note that the continuity of f(x) at a means two things:

(i)
exists,
(ii)
and this limit is f(a).
So to be discontinuous at a, means
(i)
does not exist,
(ii)
or if exists, then this limit is not equal to f(a).

Basic properties of limits imply the following:

Theorem. If f(x) and g(x) are continuous at a. Then

(1)
f(x) + g(x) is continuous at a;
(2)
is continuous at a, where is an arbitrary number;
(3)
is continuous at a;
(4)
is continuous at a, provided ;
(5)
If f(x) is positive, i.e. , then is continuous at a;
(6)
If f(x) is continuous at a and g(x) is continuous at f(a), then their composition is continuous at a.

Remark. Many functions are not defined on open intervals. In this case, we can talk about one-sided continuity. Indeed, f(x) is said to be continuous from the left at a iff

and f(x) is said to be continuous from the right at a iff

Example. The function is defined for . So we can not talk about left-continuity of f(x) at 0. But since

we conclude that f(x) is right-continuous at 0.

This concept is also important for step-functions.

Example. Consider the function

The details are left to the reader to see

and

So we have

Since f(2) = 5, then f(x) is not continuous at 2.

continuous at x=1.

Definition. For a function f(x) defined on a set S, we say that f(x) is continuous on S iff f(x) is continuous for all .

Example. We have seen that polynomial functions are continuous on the entire set of real numbers. The same result holds for the trigonometric functions and .

The following two exercises discuss a type of functions hard to visualize. But still one can study their continuity properties.

Exercise 3. Let us modify the previous function: Discuss the continuity of

for . (Two natural numbers p and q are coprime, if their greatest common divisor equals 1.)

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