Three definitions of arc length

Question

Let $\gamma:[0,1]\to M$ be a curve on a metric space $(M,d)$. There are three ways to define the arc length of $\gamma$.

1. $$\ell(\gamma):=\sup\bigg\{\sum_{i=0}^kd(\gamma(t_{i+1}),\gamma(t_i)):0=t_0<t_1<\cdots<t_k=1\bigg\}.$$
2. $$\ell(\gamma):=\lim_{\varepsilon\to0}\inf\bigg\{\sum_{i=0}^kd(\gamma(t_{i+1}),\gamma(t_i)):0=t_0<t_1<\cdots<t_k=1,|t_{i+1}-t_i|<\varepsilon\bigg\}.$$
3. If $M$ is a Riemannian manifold and $\gamma$ is differentiable, $$\ell(\gamma):=\int_0^1|\gamma'(t)|\,dt.$$

My question is when do these notions coincide. In particular:

• Are 1. and 2. the same for all continuous curves? (The length may be $\infty$.)
• Does 3. coincide with 1. or 2. if $\gamma$ is absolutely continuous? I could prove this using the mean value theorem if $\gamma$ is $C^1$, but I am not sure how to generalize this to the absolutely continuous case.

Thanks in advance!