Are all continuous everywhere but nowhere differentiable functions fractal in nature?

Question

Functions like the Weierstrass function or van der Waerden's function exhibit self-similar plots. Is this characteristic of continuous everywhere, differentiable nowhere functions? Is there a counterexample to this?

Answer 1

The Takagi function, defined by $$T(x)=\sum_{n=0}^\infty {{\rm dist}(2^nx,\mathbb Z)\over 2^n}$$ is an example of a continuous, nowhere differentiable function whose graph is not a fractal, in the sense that it has Hausdorff and box-counting dimension 1 (see here and here).

I'm sure various things in this direction are true. For example (Corollary 11.2 in Falconer's Fractal Geometry), the graph of a continuous function $f$ on $[0,1]$ that satisfies a Hölder condition with exponent $0\le\alpha\le 1$ has Hausdorff and box-counting dimensions less than or equal to $2-\alpha$.
If instead there exists $c>0$, $\delta_0>0$, and $1\le s<2$ such that for all $0<\delta\le\delta_0$ there exists $y$ with $|x-y|\le\delta$ and $|f(x)-f(y)|\ge c\delta^{2-s}$, then the box-counting dimension is at least $s$. (Note that this doesn't say anything about Hausdorff dimension, which is never larger than box-counting dimension).