Functions with distant fibers and uniform continuity

Abstract

The uniformly approachable functions introduced in [Quaestiones Math. 18 (1995) 381–396] are defined by a property stronger than continuity and weaker than uniform continuity, which is preserved under composition. (So they give rise to a category which sits between the category of metric spaces with all continuous functions and the category of metric spaces with all uniformly continuous functions.) Solving a problem left open in [Rend. Istit. Mat. Univ. Trieste 25 (1993) 23–56], we give a complete characterization of the polynomial maps f : R n→ R which are uniformly approachable. They coincide with the polynomial maps f with distant fibers, i.e., such that any two distinct fibers f −1( x) and f −1( y) are at positive distance. The same holds more generally for any real valued function on R n whose fibers have finitely many connected components. To prove this we show that every real valued continuous function with distant fibers on a uniformly locally connected metric space is uniformly approachable, and any (weakly) uniformly approachable function on R n has “distant connected components of fibers”. We observe that a bounded continuous function f : R n→ R has distant fibers if and only if it is uniformly continuous. This suggests that for a reasonable metric space X the uniform continuity of a bounded continuous function f :X→ R depends only on the fibers of f. We show that this is the case when X is connected and locally connected. A useful tool in the study of uniformly approachable functions on domains more general than R n is given by the technique of “truncations” ( g is a truncation of f if it is locally constant where it differs from f). On R n the functions with many uniformly continuous truncations coincide with the functions with distant connected components of fibers. We improve the technique of the magic set introduced in [Rend. Istit. Mat. Univ. Trieste 25 (1993) 23–56] and studied by M.R. Burke and K. Ciesielski showing that every continuous function with “small fibers” on a locally arcwise connected metric space X has a magic set M⊂ X (i.e., every continuous g :X→ R with g( M)⊂ f( M) is a truncation of f).