On the Continuity of the Modulus of Continuity
by
Joze Malesic
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Coauthors: Dusan Repovs (Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia)

Using one of Michael's classical theorems on existence of continuous selections for multivalued mappings, Repovs and Semenov proved the following result from real analysis: Let (X, d) and (Y, \rho) be metric spaces and suppose that X is locally compact. Let C(X, Y) be the space of all continuous maps from X to Y, endowed with the usual topology of uniform convergence. Then there exists a continuous singlevalued function \delta: C(X, Y) ×X ×(0, \infty) --> (0, \infty) such that for every (f, x, \epsilon) in C(X, Y)×X×(0, \infty) and for every x' in X : d(x, x') < \delta(f, x, \epsilon) ===> \rho(f(x), f(x')) < \epsilon.

Alternatively, this result can be proved using a classical theorem of Dowker on continuous separation of a lower and upper semicontinuous functions. As a corollary, one obtains an elementary proof that the Cantor theorem on uniform continuity implies the Weierstrass theorem on boundedness of continuous functions on compacta. We shall present subsequent joint work with Repovs on the modulus of continuity and its generalizations.

Date received: June 21, 2000