Hausdorff–Besicovitch dimension of the graph of one continuous nowhere-differentiable function - O., B. Panasenko (Ukrains’kyi Matematychnyi Zhurnal, 2009)
We investigate fractal properties of the graph of the function$$y = f(x) = ∑^{∞}_{k−1}\frac{β_k}{2^k} ≡ Δ^2_{β_1β_2…β_k…},$$where$$\beta_1 = \begin{cases} 0 & \mbox{if } \alpha_1(x) = 0,\\ 1 & \mbox{if } \alpha_1(x) eq 0,\\\end{cases}$$$$\beta_k = \begin{cases} β_{k−1} & \mbox{if } \alpha_k(x) = \alpha_{k-1}(x),\\ 1 - β_{k−1} & \mbox{if } \alpha_k(x) eq \alpha_{k-1}(x),\\\end{cases}$$and $‎α_k(x)$ is the kth ternary digit of $x$: In particular, we prove that this graph is a fractal set with Hausdorff–Besicovitch $α_0(Г_f) = \log_2(1 +2^{\log_32}$ dimension and cell dimension $α_K (Г_f) = 2-\log_32$.