Definition:Differentiability Class/Domain Restriction

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

Let $S \subseteq \R$ be a subset of $\R$ on which the $n$th derivative of $f$ is continuous on $S$.

Then $\map f x$ is of differentiability class $C^n$ on $S$.

Motivation
By selecting a specific domain on which to restrict a 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.