How is the set $C(f)\cap V$ of second category in $V$?

Question

I am reading the paper "P. S. Kenderov, I. S. Kortezov and W. B. Moors, Continuity points of quasi-continuous mappings, Topology Appl. 109 (2001), 321–346." Just before Theorem 2 of the paper, the authors of the paper state: "The set of points of continuity $C(f)$ is not necessarily residual in $Z$. It is however of the second Baire category in every non-empty open subset of $Z$. That is, for every non-empty open subset $V \subset Z$ the set $C(f )\cap V$ is not of the first Baire category." They have proved this in the theorem, but I did not understand it. I can only see that there exists a first category set $H$ such that for any open set $V$, $C(f )\cap V \subset V \setminus H$. What am I missing?