SOME INTERESTING SMALL SUBCLASSES OF THE DARBOUX BAIRE 1 FUNCTIONS

Abstract

We show that in the space, $\mathcal{D}{B}_1$, of Darboux Baire $1$ functions on the unit interval, equipped with the metric of uniform convergence, the subspace $\mathcal{G}$, consisting of those functions whose graph restricted to the points of continuity is dense in the entire graph, is closed and nowhere dense, and that the subspace, $\mathcal{D}{B}_1^{*}$, of Darboux Baire* $1$ functions is first category relative to $\mathcal{G}$; or stated in an equivalent manner, the space of universally first return continuous functions is a closed nowhere dense subspace of the space of first return continuous functions, and that $\mathcal{D}{B}_1^{*}$ is first category in the space of universally first return continuous functions.