Lebesgue's Differentiation Theorem for Continuous Functions

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 $$ \lim_{r\to 0}\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y)\;dy=f(x)$$