There are two completely different "representation theorems" (one famous, one not so famous) for functions of bounded variation.  They yield the same theory and are ultimately connected (via the below centred equation), though this connection is not obvious at first glance.  Here is the short version:

On one hand, the most basic result for functions $f:[0,1]\rightarrow \mathbb{R}$ of bounded variation is the *Jordan decomposition theorem* (see [here][1]), namely that $f=g-h$ where $g, h$ are monotone functions, one of which can be taken to be $\lambda x.V_0^x(f)$, where the latter is the variation of $f$ on $[0,x]$ for $x\in (0,1]$.

On the other hand, Banach proves the following surprising result (for continuous functions, but it can be generalised to any function of bounded variation): 
$$
V_{a}^{b}(f)=\int_{\mathbb{R}} N(f)(y) dy, \text{ where $N(f)(y)=\# \{x\in [a,b ]: f(x)=y\}$},
$$
where $N(f)$ is the *Banach indicatrix*.  To define $N(f)$ for discontinuous $f$ of bounded variation, one uses Sierpinski's decomposition theorem, implying that for $f$ of bounded variation, there is continuous $g$ and strictly increasing $h$ with $f=g\circ h$.  One can show that $N(g)=N(f)$ for such functions.




  [1]: https://encyclopediaofmath.org/wiki/Function_of_bounded_variation#Jordan_decomposition