Continuity set of a function
= 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 $x$ by
$$
\omega_f(x)=\inf_{r>0}\sup\{|f(y)-f(z)|:y,z\in(x-r,x+r)\}.
$$
The sets $\{x:\omega_f(x)<\varepsilon\}$ are open, and $f$ is continuous at $x$ exactly when $\omega_f(x)=0$. Hence its continuity set is the $G_\delta$ <Borel set>
$$
\bigcap_{n=1}^{\infty}\{x:\omega_f(x)<1/n\}.
$$