ON CONTINUOUS CHOICE IN THE DEFINITION OF CONTINUITY

by

D.Repovš, Ljubljana, Slovenia and P.V.Semenov, Moscow, Russia

Abstract. Using a classical theorem of C.H.Dowker on continuous separation of lower and upper semicontinuous functions, we prove the following result: Let (X, d) and (Y, ρ) be metric spaces and suppose that X is locally compact. Then there exists a continuous function δ̂: C(X, Y) × X × (0, ∞) → (0, ∞) such that for every (f, x, ϵ) ∈ C(X, Y) × X × (0, ∞) and for every x′ ∈ X the following implication holds: d(x, x′) < δ̂(f, x, ϵ) ⟹ ρ(f(x), f(x′)) < ϵ. As a corollary, we obtain that the Cantor theorem on uniform continuity follows from the Weierstrass theorem on boundedness of continuous functions on compacta. Mathematics subject classification: Primary: 54C65, 54E45 Secondary: 26A15, 52A05. Keywords: local compactness, paracompactness, metric space, lower and uppersemicontinuous function, separation by continuous function.