Show that $F_h(x)=\frac{1}{ C h^d} \int_{B(x, h)} f(y) d y$ is continuous

Question

In $\mathbb{R}^d$ provided with a norm, we note $C=\lambda(B(0,1))$ the Lebesgue measure of the unit ball. For $f \in L_{\text {loc }}^1\left(\mathbb{R}^d\right)$ and $h>0$, we consider $$F_h(x)=\frac{1}{ C h^d} \int_{B(x, h)} f(y) d y.$$ I need to show that $F_h$ is continuous ! By a variable change I have showed that $F_h(x)=\frac{1}{C}\int_{B(0,1)}f(x+hz)dz$ but I get stuck here ! I think I should use continuity theorem but $f$ is not continuous ! Any help is highly appreciated !