I wonder that whether there exist a nowhere differentable continuous function with its graph in $\mathbb{R}^2$ has Hausdorff dimension $1$.
A result about Weierstrass's function is that $\sum_{k=1}^{\infty}a^{(s-2)k}\cos a^kx$ has Hausdorff dimension $s$, provided $1<s<2$. But I don't know more about the case $s=1$.
The Takagi function (see [1]), rediscovered by Van der Waerden, is nowhere differentiable (see [1] or [2]). The Hausdorff dimension of its graph is 1 (see [3]). More information is in the survey [4].
[1] T. Takagi, A simple example of the continuous function without derivative, Phys.-Math. Soc. Japan 1 (1903), 176-177. The Collected Papers of Teiji Takagi, S. Kuroda, Ed., Iwanami (1973), 5–6.
[2] P. Billingsley, Van Der Waerden’s Continuous Nowhere Differentiable Function, Amer. Math. Monthly 89 (1982), no. 9, 691.
[3] R. Mauldin and S. Williams, On the Hausdorff dimension of some graphs, Trans. Amer. Math. Soc. 298 (1986), 793–803.
[4] Allaart, Pieter C.; Kawamura, Kiko (11 October 2011), The Takagi function: a survey, arXiv:1110.1691