In one dimension, a function $f: I:= [a,b] \to \mathbb{R}$ is said to be of bounded variation if $$\mathrm{Var}(f,I):= \sup_{P}\left \{ \sum_{i=1} ^n \|f(x_i) - f(x_{i-1})\| : \mathrm{for} \; P:=\{x_0,\dots, x_n\} \; \mathrm{partition \; of \;} I \right \} < \infty.$$
In many variables, we have that a function $u \in L_{loc}^1(\Omega, \mathbb{R})$ is said to be of bounded variation if
$$\mathrm{Var}(u,\Omega) := \sup\left\{ \int_{\Omega} u\,\mathrm{div}\varphi \, dx:\varphi\in C_c^1(\Omega,\mathbb{R}^N), \|\varphi\|_{\infty} \leq 1 \right\} < \infty.$$ Where $\Omega \subset \mathbb{R}^N$ is a bounded domain (open and connected in $\mathbb{R}^N$).
Question. In the case of one variable, if we set $\Omega = I$ and $u = f$. How are these definitions equivalent?
My thoughts:
Since $\varphi\in C_c^1(\Omega,\mathbb{R}^N)$, then $\varphi$ is of bounded variation. then use Riemann Stieltjes integral, but I may need my function $u$ to be in $(C(I), \|\cdot\|_{\infty})$. Is $u$ in $(C(I), \|\cdot\|_{\infty})$ to write $\int_{\Omega} u \, d\varphi$?
Second, how is supremum over the partitions equal to the supremum over functions in $C_c^1(\Omega,\mathbb{R}^N)$ with $\|\varphi\|_{\infty} \leq 1$
Many thanks