piecewise smooth

A curve $\alpha :[a,b]\to {ℝ}^n$ is said to be piecewise smooth if each component ${\alpha }_1,\mathrm{\dots },{\alpha }_n$ of $\alpha$ has a bounded derivative ${\alpha }_i^{\prime }\mathit{\hspace{1em}}(i=1,\mathrm{\dots },n)$ which is continuous everywhere in $[a,b]$ except (possibly) at a finite number of points at which left- and right-sided derivatives exist.

• Every piecewise smooth curve is rectifiable.

• Every rectifiable curve can be approximated by piecewise smooth curves.