I'm studying for a Qualifying exam and can't figure out this problem. I see that the limit must be $f(x)$ and can get the boundedness but had trouble with continuity. Any suggestions?
Let $f\in L^{\infty}(\mathbb{R}^{d})$ and let $\phi:\mathbb{R}^{d} \times (0,\infty)\rightarrow \mathbb{R}$ be the following map:
$\phi(x,r)=\frac{1}{r}\int_{B_{r}(x)}f(y)dy.$
Prove that $\phi$ is continuous in $x$, and in $r$, and is uniformly bounded. What can you say about $\lim_{r\rightarrow 0}\phi(x,r)$?