# Is every $G_\delta$ set the set of continuity points of some function $f$?

I can prove that given a function $f:X \rightarrow Y$, where $X,Y$ are metric spaces, the set $A \subseteq X$ of points on which $f$ is continuous, is $G_{\delta}$.

(Take $U_n = \bigcup_{y \in Im(f)} f^{-1}(B_{\frac{1}{n}}(y))$ and $V= \bigcap_{n \in \mathbb{N}}U_n$, and $V$ is $G_\delta$).

My question is: Is the converse direction true? Is it true that, given a $G_\delta$ set $A \subseteq X$, there exists a metric space $Y$, and a function $f:X \rightarrow Y$, such that, the set of points on which $f$ is continuous is $A$?

Thank you!
Shir

#### Solutions Collecting From Web of "Is every $G_\delta$ set the set of continuity points of some function $f$?"

Yes. See Theorem 2.1 on page 2 of

http://artsci.kyushu-u.ac.jp/~ssaito/eng/maths/Gdelta.pdf

In fact, every dense $G_\delta\subset\mathbb R$ is also the set of continuity of a derivative of a differentiable function!