Given $u\in W^{1,p}_{loc}(U)$, define $$u_{x,r}:=\frac{1}{|B(x,r)|}\int_{B(x,r)}u(y)dy. $$ I proved that $$ \frac{d}{dr}u_{x_0,r}\le Cr^{\frac{\varepsilon}{p}-1} $$ for $r\in (0,\frac{1}{2}\text{dist}(x_0,\partial U))$ and $\varepsilon>0$ small enough.
I don't know how to conclude that $$ \lim_{r\to 0}u_{x_0,r} $$ exists and is finite.
I thought to use Poincaré inequality, but I can't succeed.
Fix some $r_0$ so that $B(x_0,r_0)\subset U$. If you know that $(0,r_0)\ni r\mapsto u_{x_0,r}$ is differentiable and satisfies $$ \left|\frac{d}{dr}u_{x_0,r}\right|\leq Cr^{\frac{\varepsilon}{p}-1}, $$ you can use the fundamental theorem of calculus to observe that $$ u_{x_0,r}=u_{x_0,r_0}-\int_r^{r_0}\frac{d}{ds}u_{x_0,s}ds $$ for all $r\in(0,r_0)$ and the limit as $r\to0$ exists and is finite since the integral is absolutely convergent (by the estimate).