Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a given continuous function and $t_0\in (a,b)$ a fixed point. Is it true that the following problem has a unique continuous solution $\mathbf{r}:(a,b)\to\mathbb{R}^2$:

$\left\{\begin{array}{ll}\mathbf{v}(t)=\mathbf{r}(t)+\displaystyle\lim_{s\searrow t} \frac{\mathbf{r}(s)-\mathbf{r}(t)}{||\mathbf{r}(s)-\mathbf{r}(t)||}\\ \mathbf{r}(t_0)=\mathbf{v}(t_0)+(\cos\theta_0,\sin\theta_0)\end{array}\right.$ ?

sandtare real variables. Isn't that it was a typo for $\|\mathbf{r}(s)-\mathbf{r}(t)\|$? $\endgroup$ – Pietro Majer Nov 11 '14 at 23:14