# Definition:Differentiability Class

## Contents

## Definition

Let $f: \R \to \R$ be a real function.

Then $f \left({x}\right)$ is of **differentiability class** $C^k$ if and only if $\dfrac {\mathrm d^k} {\mathrm d x^k} f \left({x}\right)$ is continuous.

That, $f$ is in **differentiability class $k$** if and only if there exists a $k$th derivative of $f$ which is continuous.

If $\dfrac {\mathrm d^k} {\mathrm d x^k} f \left({x}\right)$ is continuous for all $k \in \N$, then $f \left({x}\right)$ is of **differentiability class** $C^\infty$.

## Specific Instances

**Differentiability class** $C^0$ consists of the set of continuous real functions whether they be differentiable or not.

### Continuously Differentiable

A differentiable function $f$ is **continuously differentiable** if and only if $f$ is of differentiability class $C^1$.

That is, if the first order derivative of $f$ (and possibly higher) is continuous.

### Smooth Function

A real function is **smooth** if it is of differentiability class $C^\infty$.

That is, if it admits of continuous derivatives of all orders.

## Domain Restriction

By selecting specific domains on which to restrict a given function, points at which a derivative for a given order is not continuous can be deliberately excluded.

Hence it can often be specified that a given function be smooth, for example, on a particular real interval.

## Also known as

A real function in **differentiability class** $C^n$ can be described as being **$n$ times differentiable**.