Question: If D is the set of discontinuities of $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$, show that the set of discontinuities of $f_{\Phi_{A}}$ is contained in $D \bigcup \partial A$.
So, $f_{\Phi_{A}}$ is the characteristic function of $f$, which is \begin{cases} 1 & if x\in A \\ 0 & if x\notin A \\ \end{cases} Then, pick a point $p\in f_{\Phi_{A}}$ and show that $p$ is contained in $D \bigcup \partial A$. Is this the right way to prove this? Any hints would be appreciated. Thank you.
What you have to show is $f\Phi $ is continuous at $x$ if $x \notin D\cup \partial A$. So assume $x \notin D\cup \partial A$ and take $x_n \to x$. Since $x \notin \partial A$, either $x$ is in the interior of $A$ or in the exterior. In the first case $x_n$ is also in the interior of $A$ (hence in $A$) for $n$ sufficiently large. Hence $(f\Phi )(x_n)=f (x_n) \to f(x) =(f\Phi ) (x)$ because $x \notin D$. Similar argument in the second case.