I find this pretty hard and it would be awesome if someone could help me.
The problem is the following (Problem 6/Chapter 3 from S&S's Real Analysis).
Suppose $F$ is a bounded measurable function on $\mathbb{R}$. If $F$ satisfies either one of the two following conditions:
(a) $\int_{\mathbb{R}}{|F(x+h)-F(x)|dx} \leq A|h|$, for some constant $A$ and all $h\in \mathbb{R}$;
(b) $|\int_{\mathbb{R}}{F(x)\phi '(x) dx}| \leq A$, where $\phi$ ranges over all $C^{1}$ functions of bounded support with $\sup_{x \in \mathbb{R}}{|\phi (x)|} \leq 1$; then $F$ can be modified on a set of measure zero as to become a function of bounded variation on $\mathbb{R}$.
Moreover, on $\mathbb{R}^{d}$ we have the following assertion. Suppose that $F$ is a bounded measurable function on $\mathbb{R}^{d}$. Then, the following two conditions on $F$ are equivalent:
(a') $\int_{\mathbb{R}^d}{|F(x+h)-F(x)|dx} \leq A|h|$, for some constant $A$ and all $h\in \mathbb{R}^d$;
(b') $|\int_{\mathbb{R}^d}{F(x)\frac{\partial {\phi}}{\partial{x_{j}}} dx}| \leq A$, for all $j=1,\ldots, d$, for all $\phi \in C^{1}$ of bounded support with $\sup_{x \in \mathbb{R}^d}{|\phi(x)|} \leq 1$.
I proved already that if $F$ is a BV function then (a) and (b) hold; so the first part should be a converse for that..