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A 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 $\alpha_{i}' (i = 1, \ldots, n)$ which is continuous everywhere in $[a, b]$ except (possibly) at a finite number of points at which left- and right-sided derivatives exist.
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A curve $\alpha : [a, b] \rightarrow \mathbb{R}^{k}$ is said to be \textbf{piecewise smooth} if each component $\alpha_{1}, ..., \alpha_{k}$ of $\alpha$ has a bounded derivative $\alpha_{i}' (i = 1, ..., k)$ which is continuous everywhere in $[a, b]$ except (possibly) at a finite number of points at which left- and right-handed derivatives exist.
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