Definition:Differentiability Class

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

Then $\map f x$ is of differentiability class $C^k$ :
 * $\dfrac {\d^k} {\d x^k} \map f x \in C$

where $C$ denotes the class of continuous real functions.

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

If $\dfrac {\d^k} {\d x^k} \map f x$ is continuous for all $k \in \N$, then $\map f x$ is of differentiability class $C^\infty$.

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

Some authors use $C^{\paren n}$ for $C^n$.

Some use $\mathrm C^{\paren n}$.

Some use $\mathrm C^\omega$ for differentiability class $C^\infty$.