A curve $\alpha : [a, b] \rightarrow \mathbb{R}^{n}$ is said to be piecewise smooth if each component $\alpha_{1}, \ldots, \alpha_{n}$ of $\alpha$ has a bounded derivative $\alpha_{i}' \quad (i = 1, \ldots, n)$ 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.