I will assume $\Omega$ is an interval $[a,b]$, say. I assume this was intended as part of your question.

Let $\text{Var}f=M$. Let $r>0$. The function has at most countably many discontinuities, which are necessarily of jump type. The magnitude of the discontinuities sums to at most $M$. In particular, there are at most $M/r$ points at which the function jumps by $r$ or more. Let these be $z_1,\ldots, z_k$. Let $z_0=a$ and $z_1=b$. Then on each interval $(z_i,z_{i+1})$, there exist $z_i:=x^i_0<x^i_1<\ldots<x^i_{n_i}:=z_{i+1}$ such that $r<\text{Var}_{[x^i_j,x^i_{j+1}]}(f)<2r$ for each $i,j$, except that no lower bound is imposed for $j=n_i-1$ (that is the variation is split into chunks of size roughly $r$). Now the part of the graph lying over $[x_j^i,x_{j+1}^i]$ can be covered by approximately $\lceil (x^i_{j+1}-x^i_j)/r\rceil$ balls of radius $r$, so that approximately $N_i=(z_{i+1}-z_i)/r+n_i+1$ balls are needed to cover the section of the graph lying over $[z_i,z_{i+1}]$. Notice that $n_i\le \text{Var}_{(z_i,z_{i+1})}f/r+1$, so that summing the $N_i$'s we obtain a cover with approximate $(b-a)/r+\text{Var}_{[a,b]}f/r+k$ balls of size $r$. Since $k\le M/r$, we see the number of balls is bounded by approximately $(b-a)/r+2M/r$. Hence the Hausdorff dimension is at most 1.

essential graph, i.e. to say thatessential graphof $u$ is the essential range of the function $x\mapsto(x,u(x))$. $\endgroup$ – Skeeve Apr 11 '19 at 13:5112more comments