# Differentiable function

## Definition

### Definition at a point

Consider a function $f$ of one variable. We say that $f$ is differentiable at a point $c \in \R$ 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