TITLE: Lebesgue's Differentiation Theorem for Continuous Functions
QUESTION [6 upvotes]: If $f:\mathbb{R}^n\to\mathbb{R}$ is continuous, does Lebesgue's differentiation theorem hold at all points? That is, does $$\lim_{r\to0}\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y) \, dy=f(x)$$ $\textit{everywhere}$?

REPLY [12 votes]: Yes. Fix $x\in\mathbb{R}^n$ and $\varepsilon>0$, and choose $\delta>0$ such that if $|x-y|<\delta$ then $|f(x)-f(y)|\leq\varepsilon$. If $0<r<\delta$, then
$$\Big|\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y)\;dy-f(x)\Big|\leq \frac{1}{B(x,r)}\int_{B(x,r)}|f(x)-f(y)|\;dy\leq\varepsilon$$
which shows that
$$ \lim_{r\to 0}\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y)\;dy=f(x)$$