Differentiable function

Definition

Definition at a point

Consider a function $f$ of one variable. We say that $f$ is differentiable at a point $c\in \mathbb {R}$ if the derivative of $f$ at $c$ exists (as a finite number), i.e., we get a finite limit for the difference quotient.

Note that for a function to be differentiable at a point, the function must be defined on an open interval containing the point.

Definition on an open interval

Suppose $I$ is an interval that is open but possibly infinite in one or both directions (i.e., an interval of the form $(a,b),(a,\infty ),(-\infty ,b),(-\infty ,\infty )$). We say that $f$ is differentiable on $I$ if $f$ is differentiable at every point of $I$.

Definition on a union of open intervals

A function is differentiable on a union of open intervals if it is differentiable on each of the open intervals.

Definition without specification

When we say that a function is differentiable (without specifying a point or interval or union of intervals), we mean that it is differentiable on its entire domain of definition. This makes sense if the domain is a union of open intervals.

For functions that are defined for all real numbers, this means that the function is everywhere differentiable, i.e., differentiable for all real numbers.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
continuously differentiable function	differentiable and the derivative is a continuous function	(by definition)	differentiable not implies continuously differentiable
multiply differentiable function ($k$ times differentiable for some positive integer $k$)	the function can be differentiated $k$ times
infinitely differentiable function

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
continuous function		differentiable implies continuous	continuous not implies differentiable