Continuous Function

analysis

Definition

Continuous Function

A function is called continuous at if the limit of the function as approaches exists and is equal to the function value at that point:

The function is called continuous on if it is continuous for all .

Epsilon-Delta Definition

Alternatively, is continuous at if for every , there exists a such that for all :

Sequential Continuity

Continuity is equivalent to sequential continuity: is continuous if and only if for every sequence :

Properties

Existence of a Delta Neighbourhood

Existence of a Delta Neighbourhood

For every continuous function with , there exists a delta neighbourhood such that for all . Analogously, the same applies for .

Existence of a Delta Neighbourhood

Let be continuous and . By the definition of continuity, there exists a such that for all :

For , this implies .

Preservation of Sign

For every continuous function with , there exists a delta neighbourhood such that for all . For the case , an analogous statement holds.

Image of a Closed Interval

Let be a closed interval and a continuous function. Then is also a closed interval. (See Extreme Value Theorem).

Interval on Strictly Monotone Continuous Function

Let be an interval and be a strictly monotonic continuous function. Then there exists an inverse function that is also continuous.