Let $C$ be a Cantor set on the real line. Define the Cantor-Lebesgue function $F_C$ on this set, and let the collection of sets $\{F_C \}$ be such that every function $F \in \{ F_C \}$ is of the form $F_C +k$ where $k \in \mathbb{R}.$
I can kind of see how the union $\bigcup_C \{F_C \}$ would be dense in the set of all continuous real-valued functions on $\mathbb{R}$ (but if this intuition is wrong please correct me), but I cannot see at all about proving that this is $G_\delta$.
I want to use that this union contains a dense $G_\delta$ set to prove that the typical continuous function does not map a first category set $\subset [0,1]$ to a first category set.
EDIT: By "a cantor set" I mean a compact, perfect set with empty interior.