It is of course well-known that there are plenty of functions from $\mathbb R$ into itself which are continuous and nowhere differentiable. Although the Baire Category Theorem is enough to prove the existence (and genericity) of such functions, it is also possible (and instructive) to construct directly some examples: my favorite example is given in a paper by John McCarthy, An everywhere continuous nowhere differentiable function, Amer. Math. Monthly 60 (1953), 709 [MR 0057955]. The main reason for which I believe that the McCarthy example is the best is that it is possible to describe it completely in less than one page.
$\bullet$ Q1 $\bullet$ My first question is the following one: let $f$ be the function constructed in the reference above. Did anybody ever calculate the Hausdorff dimension of the graph of $f$, i.e. of the set $$ C_f=\{(x,f(x))\}_{x\in \mathbb R}\subset \mathbb R^2\quad? \tag{$\sharp$}$$ I would guess that $ \mathcal H^\alpha(C_f) $ is positive and finite for some $\alpha\in (1,2)$, but I do not know that $\alpha$. As a final remark, I would say that the intuitive descriptions of continuity, yielding sentences like "a function is continuous if you can draw its graph without lifting your pencil from the sheet" looks rather inappropriate in view of this $\mathcal H^1(C_f)=+\infty$.
$\bullet$ Q2 $\bullet$ I have a second question. Let $g$ be a continuous nowhere differentiable real-valued function defined on $\mathbb R$ and let $C_g$ be defined by ($\sharp$). Is it true that $ \mathcal H^1(C_g)=+\infty $?
