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. 



Hausdorff dimension of the graph of an increasing function

> 
> **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. 


