Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation.


**Question 1.** 


How can we prove that the Hausdorff dimension of the [essential graph](https://mathoverflow.net/questions/327698/hausdorff-dimension-of-the-graph-of-a-bv-function-in-1-dimensional-setting?noredirect=1#comment818424_327698) of $u$ equal to $1$? 


**Question 2.** 


Is Question 1 equivalent to asking the following? 


*How can we prove that there exists a representative $\tilde u$ of $u$ such that the Hausdorff dimension of $\tilde u$ is equal to 1.*




---


**Note.**
I've asked a more general question [Hausdorff dimension of the graph of a BV function](https://mathoverflow.net/questions/327267/hausdorff-dimension-of-the-graph-of-a-bv-function/327310?noredirect=1#comment817269_327310), which has received a very nice (partial) answer, however, I'd like to see a simpler proof in the 1-dimensional case.



Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)
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. 


