Consider the following problem which appears in N.L.Carothers.
Let $g:\mathbb {R\to R}$ be a function defined by $g(x)=1$ if $x\in \Delta$ and $g(x)=0$ otherwise,then find the points of continuity of $g$.
Proof: If $c\in \Delta$,then $g(c)=1$,but for each $n\in \mathbb N$ there exists $x_n\notin \Delta$ and $|x_n-c|<1/n$ because $N(c,1/n)$ cannot be contained in $\Delta$.So,$g(x_n)=0$,but $(x_n)\to c$.So,$g(x_n){\large\not\to} g(c)$.Hence $g$ is not continuous at any point of the Cantor set.Now,if $c\notin \Delta$,then $g(c)=0$.Let $\epsilon>0$.Now,$\Delta$ is closed and hence $\mathbb R-\Delta\ni c$ is open and hence we can find $\delta>0$ such that $N(c,\delta)\subset \mathbb R-\Delta$.So,for each $x\in N(c,\delta)$ we have $g(x)=0$ and hence $|g(x)-g(c)|=|0-0|=0<\epsilon$,and hence $g$ is continuous at any point $\notin \Delta$.
Is my solution alright and is there any alternative way to do this proof?