Theorem 1. The Hausdorff dimension of the graph $\Gamma_f$ of $f$ equals $1$.
Proof.
Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph by boxes of size $1/n\times (f((k+1)/n)-f(k/n))$, $k=0,1,\ldots,n-1$ located above the intervals of partition. The diameters of each of the boxes is bounded by
$1/n+(f((k+1)/n)-f(k/n))$ and the sum of diameters is bounded by
$$
\sum_{n=0}^{n-1} \frac{1}{n}+\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)\right)=2.
$$
Since the diameters of the boxes covering the graph converge to $0$ as $n\to\infty$, it follows that the Hausdorff $\mathcal{H}^1$ measure of the graph is bounded by $2$.
$\Box$
Note that the the argument used in the proof shows also that the length of the graph is bounded by $L(\Gamma_f)\leq 2$. The graph of $f$ can be parametrized by a curve $\gamma:[0,1]\to\mathbb{R}^2$, $\gamma(t)=(t,f(t))$.
In fact a stronger result is true:
Theorem 2. $\mathcal{H}^1(\Gamma_f)=L(\Gamma_f)\leq 2$.
This is a consequence of a more general results about curves in metric spaces that we will discuss now.
If $\gamma:[a,b]\to (X,d)$ is a curve in a metric space, then its length is defined by
$$
L(\gamma)={\rm Var}(\gamma)=
\sup\left\{\sum_{i=1}^{n-1}d(\gamma(t_i),\gamma(t_{i+1}))\right\}
$$
where the supremum is over all $n$ and all partitions
$a=t_1<\ldots<t_n=b$.
$\gamma$ is said to be rectifiable if $L(\gamma)<\infty$.
Theorem 3. Every rectifiable curve $\gamma:[a,b]\to (X,d)$ can be reparametrized as a $1$-Lipschitz curve.
See Theorem 4.2.1 in [1]. This is so called arc-length parametrization.
Theorem 2 is a straightforward consequence of the following more general result and the fact that $L(\Gamma_f)\leq 2$.
Theorem 4. If $\gamma:[a,b]\to (X,d)$ is an injective rectifiable curve and $\Gamma=f([a,b])$, then
$$
\mathcal{H}^1(\Gamma)=L(\gamma).
$$
Proof. By Theorem 3 we can assume that $\gamma$ is Lipschitz and in the Lipschitz case Theorem 4 is proved is Theorem 4.4.2 in [1].
$\Box$
[1] L. Ambrosio, P. Tilli, Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford, 2004.
