Translation in $L^{\infty}$

Consider the translation operator $\tau_h$ defined on $L^\infty(\mathbb{R}^n)$ s.t. $\tau_hu(x)=u(x-h)$. I know that $\tau_h$ is not continuous with respect to $h$, I mean it’s not true that $h\to 0$ implies $|| \tau_hu - u ||_{L^\infty}\to 0$.I should prove that the precedent statement is true iff it does exists a uniformly continuous function $v$ s.t $v=u$ a.e.I proved that if such a $v$ exists, the statement is true. Can someone help me with the other implication, please?