Let $S \subset \mathbb R$ be a subset of the reals. Define the characteristic function $f_S(x)$ of $S:$ $f_S(x)=\begin{cases} 1,& \text{if $x \in S$}; \\0,& \text{if $x \notin S$.} \end{cases}$
Let $C$ be the Cantor set. Prove that $f_C(x)$ is continuous almost everywhere on $[0,1]$.
My idea is that if a point $x_0$ is not in $\mathbb C$, there is a neighborhood of $x_0$ not in $\mathbb C$. But I don't know how to prove it.