real analysis - Differentiability of the Cantor Function

I know that the Cantor function is differentiable a.e. but I want to prove it without using the theorem about monotonic functions. I have already proved that $f'(x) = 0$ for all $x \in [0,1] \backslash \mathbb{C}$ where $\mathbb{C}$ is the Cantor set.

But I'm not sure how to go about proving that if $x \in \mathbb{C}$ then $f$ is not differentiable at $x$.

Actually, upon reflection, I think I have already proved differentiability a.e. but I would still like to know how to finish this part.

Also, the definition I am using for the function:
$$f:[0,1] \to [0,1]$$
Let $x \in [0,1]$ with ternary expansion $0.a_1a_2...$ Let $N$ be the first $n \in \mathbb{N}$ such that $a_n = 1$. If for all $n \in \mathbb{N}$, $a_n \in \{0,2\}$, let $N = \infty$.

Now define $b_n = \frac{a_n}{2}$ for all $n < N$ and $b_N = 1$.
Then $$f(x) = \sum_{i=1}^{N} \frac{b_n}{2^n}.$$