Does any curve have a constant speed parameterization? Let $(M,g)$ be a smooth, connected Riemannian manifold. Let $\gamma: [a,b]\rightarrow M$ be a piecewise smooth curve in $M$.
My question: is it always possible to find a parameterization $\phi: [c,d]\rightarrow [a,b]$ such that the curve $\gamma_1:=\gamma \circ \phi$ has a constant speed in the sense that $|\gamma_1'|_g$ is constant on $[c,d]$?
Here, a parameterization is a diffeomorphism as in Lee's book.
I knew that if $\gamma$ is regular, namely $\gamma' \neq 0$ on $[a,b]$ then the answer is positive and a parametrized curve can be made with speed of 1.
 A: Initially Lee defines the concept of a reparametrization for smooth curves, but later he generalizes it to piecewise smooth curves.

A reparametrization of a smooth curve $\gamma : [a, b] \to M$ is a curve  of the form $\tilde \gamma = \gamma \circ \varphi$, where $\varphi : [c, d] \to [a, b]$ is a diffeomorphism. More generally, if $\gamma$ is piecewise smooth, $\varphi$ is allowed to be a homeomorphism whose restriction to each subinterval $[c_{i-1}, c_i]$ is a diffeomorphism onto its image, where $c = c_0 < c_1 < \ldots < c_k = d$ is some finite subdivision of $[c, d]$.

In Riemannian Manifolds: An Introduction to Curvature he states in Exercise 6.2 :

Let $γ : [a, b] → M$ be an admissible curve. Show that there exists a unique forward reparametrization $\tilde γ : [0, l] → M$ of $γ$ such that $\tilde γ$ is a unit speed curve.

An admissible curve is a piecewise regular curve - and that is the essential point. An arbitrary piecewise smooth curve $\gamma$ can have points $t \in [c_{i-1}, c_i]$ for some subinterval such that $\gamma'(t) = 0$, and no reparameterization can achieve $\tilde γ'(t) \ne 0$ (use the chain rule).
Note that in the subdivision points $c_1, \ldots, c_{n-1}$ the curve $\gamma$ may have corners (i.e. the left-hand and right-hand derivatives of $\gamma$ do not coincide at $c_i$). Nevertheless we can reparametrize it to $\tilde γ : [0, l] → M$ such that $\tilde γ$ is a unit speed curve. This means that the unit speed curve can change its direction in $c_i$. As an example take the unit speed curve $\delta : [-1,1] \to \mathbb R^2, \delta(t) = (t/\sqrt 2, \lvert t \rvert/\sqrt 2)$.
