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:


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


 * $$(2) \quad \frac {d^{k+1}} {dx^{k+1}}$$ 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 $$\frac {d^k} {dx^k}$$ 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.