Average of VMO-function continuous

Question

Does anyone know why for given $f\in \operatorname{VMO}(S^1)$, where VMO is the room of Vanishing Mean Oscillation as in D.Sarason, Functions of VMO, the map $$I_{\varepsilon}f(x) := \frac 1 {\operatorname{vol}(B_{\varepsilon}(x))} \int\limits_{B_{\varepsilon}(x)} f(y) \, dy $$ is continuous in $x$ for all $\varepsilon$?

Thanks!

Answer 1

It's continuous because $\operatorname{VMO}\subset\operatorname{BMO}\subset\operatorname{L^1}$ and in $L^1$ $$x\mapsto \int\limits_{B_{\varepsilon}(x)}f(y)dy$$ is continuous.