Let the endpoint of a differentiable, vector-valued function a$(t):(t_a\,..t_b)\rightarrow \mathbb R$ move on a circle in an Euclidean plane. Let $t_0\in(t_a\,..t_b)$. Let $s(t)$ be the length of the arc between $t_0$ and point a$(t)$. Let there be a positive real number $\delta$ such that $\forall t:0<|t-t_0|<\delta:s(t)-s(t_0)\neq0$.
My (physics) book uses a proof which seems to imply that $s(t)$, where $t$ is time, is differentiable in respect to $t$. How to prove its differentiability?
The arclength is $s(t)=\int_{t_0}^t\vert\mathbf a'(\tau)\vert \mathrm d\tau$
So, $s(t)$ is differentiable because $s'(t)=\vert\mathbf a'(t)\vert$ exists: composition of continuous functions and the derivative of $\mathbf a(t)$, that exists.