Definition:Differentiability Class

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

Then $f \left({x}\right)$ is of differentiability class $C^k$ iff:


 * $(1): \quad \dfrac {\mathrm d^k} {\mathrm dx^k}$ is continuous;


 * $(2): \quad \dfrac {\mathrm d^{k+1}} {\mathrm dx^{k+1}}$ is not continuous.

That, is the differentiability class of $f$ is the highest order derivative of $f$ which is continuous.

If there is no such highest order derivative, that is, if $\dfrac {\mathrm d^k} {\mathrm dx^k}$ is continuous for all $k \in \N$, then $f \left({x}\right)$ is of differentiability class $C^\infty$.

Continuously Differentiable

 * $f$ is continuously differentiable if the differentiability class of $f$ is at least $1$.

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

Smooth Function

 * $f$ is smooth if $f$ is of differentiability class $C^\infty$.

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.