Denote $C$ as the Cantor set and denote $\chi_K$ as its characteristic function such that:
$ \chi_K(x)=\begin{array}{cc} \{ & \begin{array}{cc} 1 & x \in C \\ \ 0 & x \notin C \end{array} \end{array}$
Now, obviously for all $x \notin C, \chi_K(x) =0$ because $\chi_K(x) =0$ if and only if $x \notin K$, simply by definition.
For the latter statement ($\lim_{x \to x_0} \chi_K(x)=0$: this is true because each no point in the Cantor set is an interior point. In other words, $C^o=\emptyset$
Therefore, for $x \in C$, $\nexists \epsilon>0: (x-\epsilon,x+\epsilon)\subseteq C$
Any advice on how to prove this mathematically?