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 $\dfrac {\mathrm d^k} {\mathrm d x^k} f \left({x}\right)$ is continuous.

That, $f$ is in differentiability class $k$ 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 $C^\infty$.

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.