# piecewise smooth

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 $\alpha_{i}^{\prime}\quad(i=1,\ldots,n)$ 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.

Title piecewise smooth PiecewiseSmooth 2013-03-22 12:53:26 2013-03-22 12:53:26 cvalente (11260) cvalente (11260) 7 cvalente (11260) Definition msc 51N05 Rectifiable