On boundary avoiding selections and some extension theorems

A theorem of Marc Frantz about controlled continuous ex-tensions of functions inspired us to prove a general result con-cerning boundary avoiding continuous selections into Banach spaces, which has Frantz ’ theorem as a corollary. In addition, with relatively simple means we improve upon some other re-sults of Frantz involving extensions of products and of disjoint families of functions. 1. Introduction. The following two extension theorems are presented in Frantz [3]. Let I denote the interval [0, 1]. Theorem 1. Let X be a normal space, let A be a closed subset of X, and let Y0, Y1 be disjoint closed Gδ-subsets of X. If f: A → I is a continuou