Let $A$ be a subset of $\mathbb{R}^n$. Show that the characteristic function $\chi_A$ is continuous on the interior of $A$ and on $A^c$ but discontinuous on the boundary of $A$.
My attempt:
Suppose $\chi_A$ is continuous on the boundary of $A$, then for all $x$ on the boundary the limit of the function at $x$ equals the function evaluated at $x$. Therefore for $x$ on the boundary, the pre-image of the function evaluated at $x$ is an open set since the function is continuous. However, for $B_r(x)$, $r>0$ contains parts of $A$ and $A^c$, therefore the function would take on two values for a point in the domain, which contradicts the fact that $\chi_A$ is a function.