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Let $\gamma :[a,b]\to\mathbb{R}^2$ be a $C^{1} ([a,b])$ injective application. Is it true that there is another continuous parametrization $\rho:[c,d]\to\mathbb{R}^2$ such that the following two sets are equal:

$\gamma([a,b])=\rho([c,d])$,

and $\rho$ admits lateral derivatives at each point on $[c,d]$ with the property that $\rho'_+ (t)\neq (0,0), \ \rho'_{-}(t)\neq (0,0),\ \forall\ t\in [c,d]$?

P.S. I have use the following notations: $\rho'_{+}(t_0)=\lim\limits_{t\searrow t_0}\dfrac{\rho(t)-\rho(t_0)}{t-t_0}$ and $\rho'_{-}(t_0)=\lim\limits_{t\nearrow t_0}\dfrac{\rho(t)-\rho(t_0)}{t-t_0}$.

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No: $[-1,1]\ni t\mapsto \begin{pmatrix} t^3\sin(1/t) \\ t.|t|\end{pmatrix}$

Edit:

As Christian Remling pointed out, this just showed that you cannot have piecewise $C^1$ reprameterizations. But $t\mapsto\begin{pmatrix} t^3\sin(1/t) \\ t.|t|^3\end{pmatrix}$ should do it.

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