The continuity of the function

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

1. 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.

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:

2. If on the interval the function is continuous and does not turn into zero on this interval function keeps a constant sign.

Example of continuity of a function

Interval methods

3. 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 .

Example of continuity of a function

is a continuous function. If

then . Because , there is a point where .



4. A function continuous on a segment , is restricted on this segment, then there exist two numbers and that for all inequality .

The rule of finding the largest and smallest snakeb functions.

5. 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.
6. The inverse of a continuous function on the given interval, is continuous on this interval.
7. If the function has a derivative at , then it is continuous at that point.

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