Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$ with smooth boundary. For any $f \in L^1 (\Omega)$, is it true that $f \in L^\infty (\Omega)$ if and only if the following condition holds?

There exists some $C > 0$ such that for almost every $x \in \Omega$, and every open ball $B$ containing $x$, we have

$$f(x) \leq C \def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_B f.$$

*Note: Here $\def \avint{\mathop{\,\rlap{-}\!\int}\nolimits} \avint_B f$ denotes the average integral of $f$ over $B$.*