Another type of derivative, and the associated primitive

Question

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so that:

$ \mathbf{v}(t_0)=\lim\limits_{t\searrow t_0} \displaystyle \frac{\mathbf{r}(t)-\mathbf{r}(t_0)}{||\mathbf{r}(t)-\mathbf{r}(t_0)||},\forall t_0\in (a,b). $

Is it true that that the only solutions are of the form: $\mathbf{r}(t)=\displaystyle\int f(\tau)\mathbf{v}(\tau)\ \mathrm{d}\tau +\mathbf{c}$, where $f:(a,b)\to (0,\infty)$ is continuous and $\mathbf{c}=(c_1,c_2)$ is a constant?

Answer 1

At any point $t_0$ where ${\bf r}$ is differentiable and ${\bf r}' \ne \bf 0$ $${\bf v}(t_0) = \lim_{t \searrow t_0} \dfrac{{\bf r}(t) - {\bf r}(t_0)}{t-t_0} \dfrac{t - t_0}{\|{\bf r}(t) - {\bf r}(t_0)\|} = \dfrac{{\bf r}'(t_0)}{\|{{\bf r}'(t_0)}\|}$$ so that ${\bf r}'(t_0) = \|{\bf r}'(t_0)\| {\bf v}(t_0)$ wherever ${\bf r}$ is differentiable.

If ${\bf r}$ is absolutely continuous, we then have ${\bf r}(t) = {\bf r}(t_0) + \int_{t_0}^t g(\tau) {\bf v}(\tau)\; d\tau$ where $g = \|{\bf r}'\|$ is locally $L^1$.

But somewhat more generally, let $c \in (a,b)$, let $\mu$ be a continuous locally finite positive Borel measure on $(a,b)$, and let ${\bf k}$ be a constant vector. Then $$ {\bf r}(t) = {\bf k} + \int_c^t {\bf v}(\tau)\; \mu(d\tau) $$ is a solution.