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 : Rn → 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 Rn 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 Rn has “distant connected components of fibers”. We observe that a bounded continuous function f : Rn → 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 Rn is given by the technique of “truncations” (g is a truncation of f if it is locally constant where it differs from f ). On Rn 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 )
Functions with distant fibers and uniform continuity
BERARDUCCI, ALESSANDRO;
2002-01-01
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 : Rn → 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 Rn 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 Rn has “distant connected components of fibers”. We observe that a bounded continuous function f : Rn → 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 Rn is given by the technique of “truncations” (g is a truncation of f if it is locally constant where it differs from f ). On Rn 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 )I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
