# Differentiable function

## Contents

## Definition

### Definition at a point

Consider a function of one variable. We say that is **differentiable** at a point if the derivative of at 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 is an interval that is open but possibly infinite in one or both directions (i.e., an interval of the form ). We say that is differentiable on if is differentiable at every point of .

### 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 ( times differentiable for some positive integer ) | the function can be differentiated 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 |