Differentiable Function

analysis

Definition

Differentiable Function

A function is called differentiable in point , if the limit . The limit is called the derivative of at point and is named .

A function is called -times differentiable in point , if the -th derivative exists, which can be defined recursively:

If is also continuous in , then is called -times continuously differentiable in .

Continuity

Differentiability Implies Continuity

If a function is differentiable in , then is also continuous at .

Differentiability Implies Continuity

Let in be differentiable. Thus:

Linear Approximation

A function is differentiable in if and only if it is linearly approximatable:

with (where refers to the Big-O Notation).

Relative Extremum in Interior

Let be a differentiable function and a relative extremum in the interior . Then