## An Interesting Application of Classical Topology in Real Analysis

### Dusan Repovs, University of Ljubljana

**Abstract**
Using one of E.Michael's classical theorems on existence of continuous
selections for multivalued mappings, we shall prove the following
result from real analysis: Let (X, d) and (Y,h) 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 D from *C*(X, Y) x X x (0, \infty) to (0, \infty) such
that for every (f,x,e) in *C*(X, Y) x X x (0, \infty) and for
every x' in X we have d(x, x') < D(f,x,e) implies that h(f(x), f(x'))
< e. Alternatively, this result can be proved using a classical
theorem of C.H.Dowker on continuous separation of a lower and upper
semicontinuous functions. As a corollary, we immediately obtain an
elementary proof that the Cantor theorem on uniform continuity implies
the Weierstrass theorem on boundedness of continuous functions on
compacta. This was a joint work with *P.V.Semenov*. We shall
also discuss subsequent joint work with *J.Malesic* on the
modulus of continuity.