Continuity and Lebesgue measure

Question

Is $f(x) = m\left(B(0,|x|) \cap A\right)$ continuous if $A\subset \mathbb R^n$ is a measurable set? If not continuous, does it at least have the intermediate value property? My guess would be that I would either need to apply the lemma of Borel-Cantelli or approximate A by an open set $A \subset O_\epsilon$ of measure $\epsilon+m(A)> m(O_\epsilon)$ but I do not see how. Thank you.

Answer 1

The answer is yes. By Lebesgue density theorem $$ \lim_{\delta\to0}\frac{m(B(0,|x|)\cap A\cap B(x,\delta))}{m(B(x,\delta))}\leqslant 1 $$ So given $\epsilon>0$, let $m(B(x,\delta))<\epsilon$ $$ |f(x+\delta)-f(x)|\leqslant m(B(x,\delta))<\epsilon $$