Prove $\limsup_{\epsilon \to 0} \epsilon^{-\gamma p} ⨍_{B(x_0,\epsilon)} |f(x)|^p dx < \infty \implies f(x) \le C |x-x_0|^\gamma$

Question

Let $x_0 \in \mathbb{R}^N$, $p \ge 1$, $\gamma>0$, $\epsilon>0$.

How can I prove that we have $$\limsup_{\epsilon \to 0} \epsilon^{-\gamma p} ⨍_{B(x_0,\epsilon)} |f(x)|^p dx < \infty \implies f(x) \le C |x-x_0|^\gamma, $$ for some $C>0$?

Are the two actually equivalent ($\iff$)?

I guess one should use a theorem on integral average, but I cannot figure out how to proceed.