Continuity set of a function

The continuity set of a function is the set of points where it is continuous. For a real function $f$, define its oscillation at $x$ byThe sets $\{x:{\omega }_f(x)<\epsilon \}$ are open, and $f$ is continuous at $x$ exactly when ${\omega }_f(x)=0$. Hence its continuity set is the $G_{\delta }$ Borel set