I know that if $f:[a,b] \rightarrow \mathbb{R}$ continuous its graph has measure zero in $\mathbb{R^2}$ but what if $f$ is just monotonic(possibly discontinuous)?
Let us denote the set of discontinuities by $D$, I was thinking on since $D$ is at most countable we have no problem there but I do not know what $[a,b]\setminus D$ looks like. I am not sure if the graph necessarily has measure zero in this case.