$f(x)=1$; $x$ is in cantors set and $0$ when $x$ is not in cantors set...check continuity

Question

$f$ is a function defined from $[0,1]$ to $\{0,1\}$ as $f(x)=1$ when $x$ belongs to $C$ and $f(x)=0$ when $x$ does not belong to $C$,where $C$ is cantor's set.

Show that $f$ is not continuous on any point of its domain.

Answer 1

Actually I don't think this is right. For example $f$ is continuous on $\frac{1}{2}$. Indeed, for $\varepsilon:=\frac{1}{12}$ (for simplicity), $f([\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon])=\{0\}$, as $[\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon]$ do not contains any Cantor's Set's points