Let $\Omega \subset \mathbb R^d$ be a bounded connected open set with smooth boundary. Suppose $f \in L^1_{\text{loc}} (\Omega)$ is such that there exist some constants $c, C > 0$ such that for every $x \in \Omega$, and every open ball $B \subset \Omega$ containing $x$, we have

$$\frac{c}{|B|} \int_{B} f \leq f(x) \leq \frac{C}{|B|} \int_{B} f.$$

Does it follow that $f \in L^\infty (\Omega)$?