We can reparameterize a curve $r : [0,1]\to\mathbb R^n$ by its arc length $L(t)$ defined by $$s=L(t)=\int_0^t|r'(u)|du$$
If $L(t)$ has an inverse, letting $t=L^{-1}(s)$, we have $$\frac {d} {ds} r(L^{-1}(s))=r'(L^{-1}(s))(L^{-1}(s))'=\frac {r'(L^{-1}(s))} {L'(L^{-1}(s))}$$
thus $|\frac {d r(L^{-1})}{ds}|=1$ since $L'=|r'|$
However this proof needs $L$ to be inversible. So we have to add a regularity condition $|r'(t)|\neq 0$ so that $|L'|\neq 0$ i.e. inversible.
But even $L$ is not reversible, there's a way out. For examaple, if $L(t)=t^2$, we can use $L^{-1}(s)=\sqrt s$ and it works. Or if there are only finitely many points of $L(t)=0$, using Implict function theorem seems to be okay.
My question is, What is the worst situation can be? Is there any example that one can't make any reparameterization by arc length while $r$ is an rectifiable euclidean curve?