# Continuous functions

The concept of continuity of a function is intuitively clear, however, very complex.

Less formal, if the domain of the function is an interval, then the graph of the function we can draw without lifting the pencil from the paper. Therefore, for instance, the function sgn: is not continuous at , since sgn does not exist:

We can assume that the existence of the limit of a function at the point is necessary condition that a function is continuous at the same point. If we consider the following graph of the function,

we can see it is not a sufficient condition. From the graph we can see that the function has the limit at point , however, the limit is not equal to the value of the function at point ().

Therefore, the formal definition follows.

Let be an open interval and . The function is continuous at point if:

1.)

exists

and

2.)

The function is the continuous function if it is continuous at every point from the domain of the function.

Cauchy definition

Let be an open interval, and function. The function is continuous at point iff :

For example, the constant function , , is continuous at every point . Namely,

The sine function is continuous on . For proof, we will use the trigonometric identity for the difference of sines and the following inequality:

for .

Therefore, for any we have

when .

Now we have:

that is

which means that the function sine is continuous at point .

Trigonometric functions are continuous on theirs domains.

Properties of continuous functions

Let be an open interval, and continuous functions at . Then:

1.) , the function is continuous at ,

2.) the function is continuous function at ,

3.) if , , then the function is continuous function at .

The function , , and is continuous function on .

Every polynomial function defined as

, , is continuous function on .

Every rational function is continuous on its domain. Rational function is a ratio of two polynomials and , that is, a ratio of two continuous functions:

Therefore, the rational function is a continuous function on such that .

Continuity of composite functions

Let be open intervals and , functions for which is valid , that is, the function is well defined. If the function is continuous at point and function continuous at point , then the function is continuous at point .

Uniformly continuous functions

Let be an open interval and . Then the function is uniformly continuous on if:

This means that depends on , not on and .

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