## The continuity of the function at the point

**Definition: **a Function is called continuous at if , that is .

## Continuous functions on the interval

**Definition: **If a function is continuous at each point of some interval , it is called continuous on the interval .

## Continuity properties of functions

- If continuous on wtrysku individual acquires at the ends of this segment the values of different signs, then at some point this segment she takes on a value of zero.
- If on the interval the function is continuous and does not turn into zero on this interval function keeps a constant sign.
- A function , continuous on an interval takes on all intermediate values between the values of this function at the extreme points, i.e., between and .
- A function continuous on a segment , is restricted on this segment, then there exist two numbers and that for all inequality .
- The amount of the difference and the work is continuous on this interval the function is continuous on the same interval function. The quotient of two continuous functions is a continuous function at all points where the denominator is not peretolchina to zero.
- The inverse of a continuous function on the given interval, is continuous on this interval.
- If the function has a derivative at , then it is continuous at that point.

**Example of continuity of a function**

is a continuous function (polynomial)

so on the interval (0;1) there is a point where the function equals 0:

**Example of continuity of a function**

**Example of continuity of a function**

is a continuous function. If

then . Because , there is a point where .

The rule of finding the largest and smallest snakeb functions.

## Break point

**Definition: **Point break point function , if the point is not the condition that when .

### Examples of functions with break points

— break points all the integer points

— break - 0

— break - 0