\begin{definition}[Definition:Differentiability Class]
Let $f: \R \to \R$ be a real function.
Then $\map f x$ is of '''differentiability class''' $C^k$ {{iff}}:
:$\dfrac {\d^k} {\d x^k} \map f x \in C$
where $C$ denotes the class of continuous real functions.
That is, $f$ is in '''differentiability class $k$''' {{iff}} 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''' $C^\infty$.
\end{definition}