piecewise smoothPiecewiseSmooth2002-07-30 16:00:512007-06-08 03:30:28Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA curve $\alpha : [a, b] \rightarrow \mathbb{R}^{n}$ is said to be \textbf{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.
\begin{itemize}
\item Every piecewise smooth curve is rectifiable.
\item Every rectifiable curve can be approximated by piecewise smooth curves.
\end{itemize}