Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Let $\Gamma_f$ denote its graph. What can be said about the Hausdorff dimension of $\Gamma_f$? In particular, is it true that it is always 1?

If not, is there a link between $\dim_H(\Gamma_f)$ and $\dim_H(\mu)$, where $\mu$ is the measure whose distribution function is $f$? (That is, $f(x)=\mu[0,x]$.)

I would appreciate some examples if there's no general answer. Specifically, I think something has to be known when $f$ is the Minkowski question mark function, but Google wasn't much help here, unfortunately.