The Cantor function (which I will define recursively below) is Hölder continuous and the exponent there is ln(2)/ln(3).
The function is defined as the limit function of a sequence of functions, where $f_{n}$ looks as below: \begin{cases} f_{n-1}(3x/2) & 0 \leq x\leq 1/3\\ 0.5 & 1/3\leq x\leq 2/3 \\ 0.5 + f_{n-1}((3x-2)/2) & 2/3\leq x \leq 1 \end{cases}
I've proved that $f_{n}$ converges uniformly to the Cantor function and that $||f_{n} - f_{n-1}||_{\infty}$ $\leq$ $\frac{1}{2^n}$, but I have no idea where to proceed from there. Any inputs will be appreciated. My final goal, as I mentioned above, is to prove Hölder continuity with Hölder exponent ln(2)/ln(3).