Let $f: [0, 1] \to \mathbb R$ be a function of bounded variation. We say that $g$ is a $C^0$ reparametrization if $g = f \circ s$ for $s$ a continuous increasing bijection from a finite interval $I$ to $[0, 1]$.
Question: Does a continuous function of bounded variation always admit a $C^0$ reparametrization that is continuously differentiable?
Remark: Reparametrizing by the (inverse of) the variation of $f$ gives a $1$-Lipschitz function, which is at least differentiable a.e.
Best Answer
Non-constant $C^1$ functions have intervals of monotonicity: if $f'(x_0)\neq 0$ then $f$ is monotone on some interval $(x_0-\epsilon,x_0+\epsilon)$; this follows from the inverse function theorem. Evidently monotone continuous reparameterizations preserve this property. But a function of bounded variation does not have to have such intervals. See examples in the comments.