From the wikipedia page a scalar function is absolutely continuous on a closed interval iff it is differentiable almost everywhere and \begin{align*} f(x) = f(a) + \int_a^xf'(t)dt \end{align*} for all $x \in [a, b]$.
This definition of absolute continuity is very convenient for me. I'd like to have something similar for a vector function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$. Would it be correct to generalize this as \begin{align*} f(x) = f(a) + \int_0^t\dfrac{\partial f(r(s))}{\partial x}r'(s)dt \end{align*} for some curve $r(s) : \mathbb{R} \rightarrow \mathbb{R}^n$ with $r(0) = a$ and $r(t) = x$? $\partial f / \partial x$ is the $m\times n$ jacobian. I imagine that this integral would have to hold for any possible curve $r$.