Let $f$ be a continuous, nowhere differentiable function defined on a closed real interval $\mathbb{I}$.
Fix $\varepsilon>0$. Applying Vitali covering lemma, you can pass to a closed subset $S\subset\mathbb{I}$ formed by a finite collection of closed intervals such that $$\lambda(f(S))>(1-\varepsilon)\cdot\lambda(f(\mathbb{I}))\quad\text{and}\quad \lambda(S)<\tfrac12\cdot \lambda(\mathbb{I}),$$ where $\lambda$ denotes Lebesgue measure.
It remains to iterate this construction for a sequence $\varepsilon_n\to 0$ such that $$\prod_n(1-\varepsilon_n)>0.$$