Is there a description of the points of the Cantor set on which the Cantor function is differentiable?

Question

Let $C$ be the usual ternary cantor set, and $f$ the Cantor function, or Devil’s staircase associated to it. We know that $f$ is differentiable a.e., and on every point of the complement $C^c$, the derivative is $0$. Is there a description of the set of points of $C$ on which $f$ is differentiable? And can we identify the derivative?

Answer 1

We can characterize the Cantor set as the set of $x \in [0,1]$ which have a base-$3$ expansion $x = \sum_{i=1}^\infty x_i 3^{-i}$ where all $x_i \in \{0,2\}$. The Cantor function on $C$ is then $f(x) = \sum_{i=1}^\infty (x_i/2) 2^{-i}$. Now if $x \in C$, you can get $y \in C$ with $|x - y| = 2 \cdot 3^{-k}$ by flipping the $k$'th digit: $y_i = x_i$ except $y_k = 2-x_i$; and $|f(x) - f(y)| = 2^{-k}$. Since $\dfrac{2^{-k}}{2 \cdot 3^{-k}} \to \infty$ as $k \to \infty$, $f$ is never differentiable at any $x\in C$.