The Covering Principle for Darboux Baire 1 functions
We show that the Covering Principle known for continuous maps of the real line also holds for functions whose graph is a connected $G_\delta $ subset of the plane. As an application we find an example of an approximately continuous (hence Darboux Baire 1) function $f: [0,1]\to [0,1]$ such that any closed subset of $[0,1]$ can be translated so as to become an $\omega $-limit set of $f$. This solves a problem posed by Bruckner, Ceder and Pearson [Real Anal. Exchange 15 (1989/90)].