Let $c : [a,b] \to \mathbb{R}^n$ be a continuous and injective curve. Set, for $[x, y] \subseteq [a,b]$,
$$ \ell(c;[x,y]) = \sup \left\lbrace \sum_{i=1}^N \Vert c(t_i) - c(t_{i-1})\Vert : x = t_0 < \cdots < t_N = y,\, N \in \mathbb{N} \right\rbrace$$
the length of $c|_{[x,y]}$. Let $L = \ell(c; [a,b])$. Define a function $\varphi : [a.b] \to [0, L]$ by $\varphi(t) = \ell(c; [a,t])$. Since $c$ is injective, $\varphi$ is strictly increasing, and hence invertible. Let $\psi : [0, L] \to [a,b]$ be the inverse of $\varphi$ and define $\tilde{c} = c\circ \psi$. How do we show that $\tilde{c}$ is parametrized by arc-length, i.e.:
$$\ell(\tilde{c}; [s,t]) = \vert t - s \vert, \quad [t, s] \subseteq [0, L]?$$
I can do it whenever $c$ is $C^1$, but how to show it in the case of a merely continuous curve?
Added: as Lee Mosher observed, it can happen that $L = \infty$, in which case $c$ is non-rectifiable. So, I also add the hypothesis that $L < \infty$.
If the quantity $$\sum_{i=1}^N \Vert c(t_i) - c(t_{i-1})\Vert : x = t_0 < \cdots < t_N = y,\, N \in \mathbb{N} $$ is not bounded above then its supremum does not exist in the real numbers, and so the quantity $\ell(c;[x,y])$ is not defined in the real numbers.
And this can happen. If it happens then the restricted curve $c \mid [x,y]$ is said to be nonrectifiable.
This may actually happen for every choice of $x,y$ such that $a \le x < y \le b$, in which case the curve $c$ is said to be nowhere rectifiable.
You can find constructions of continuous, nowhere rectifiable curves in many analysis books and/or topology books. I know there's one in Munkres "Topology", for example. For such curves, no arc length parameterization exists.