Piecewise c one curve on a manifold

A piecewise $C^1$ curve in $M$ is a continuous map $\gamma :[a,b]\to M$ with a finite subdivision such that, in local charts on each closed piece, its coordinate representative is $C^1$ on the interior and its derivative extends continuously to both endpoints. The endpoint values of this extension are the corresponding one-sided derivatives.

Use the chartwise regularity convention of $C^r$ and smooth maps between smooth manifolds and the finite path operations of Reversal, concatenation, closed paths, and oriented piecewise-C1 reparametrizations. Refining a piece into finitely many chart pieces is allowed. No nonzero-velocity hypothesis is imposed: constant segments and pauses are admissible. A singleton parameter interval is interpreted as a constant curve.

Lee, Chapter 13, pp.337–340, Proposition 13.25, Lemma 13.28 and Theorem 13.29; finite piecewise $C^1$ refinements and pauses are treated explicitly here.