Given a first category set $A \in [0,1]$, is the set $X = \{f \in C[0,1]:f(A)\text{ is first category}\}$ residual in $C[0,1]$?
I tried two strategies: First I tried to play the Banach-Mazur game on $X^c$, but it does not seem clear to me who would have a winning strategy. Secondly, I tried finding a dense $G_\delta$ set contained in X, but it did not get me very far either. How do I approach this problem?