Let $\mathbf{r}:I\subseteq\mathbb{R}\to\mathbb{R}^2$ be a $C^{1}(I)$ function, non-constant on any subinterval, such that $\forall\ t_0\in I$, the following limits exist:

$\lim\limits_{t\nearrow t_0}\dfrac{\mathbf{r}(t)-\mathbf{r}(t_0)}{||\mathbf{r}(t)-\mathbf{r}(t_0)||}$ , $\lim\limits_{t\searrow t_0}\dfrac{\mathbf{r}(t)-\mathbf{r}(t_0)}{||\mathbf{r}(t)-\mathbf{r}(t_0)||}$.

Prove that there is a function $\tilde{\mathbf{r}}:J\subseteq\mathbb{R}\to\mathbb{R}^2$, $J\subseteq\mathbb{R}$ an interval, such that the following two limits exist:

$\lim\limits_{t\nearrow t_0} \dfrac{\tilde{\mathbf{r}}(t)-\tilde{\mathbf{r}}(t_0)}{t-t_0}$, $\lim\limits_{t\searrow t_0} \dfrac{\tilde{\mathbf{r}}(t)-\tilde{\mathbf{r}}(t_0)}{t-t_0}$ and are different from $(0,0)$. Moreover, $\mathbf{r}(I)=\tilde{\mathbf{r}}(J)$.

I'm trying to obtain a connection between curves that admit lateral tangents at any point and curves that can be parametrized by arc length (those having non-vanishing derivative). I think that it is enough to consider arc length parametrized curves to model the bicycle tracks. I have reached to this assertion after more failed attempts (like the following one: see Image of any curve can be parametrized without zero derivative). Now I hope that it's alright. Thanks for understanding and help!


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