This paper explores some of the
possibilities for controlling properties of continuous extensions of continuous
functions. For example, particular attention is paid to the problem of preserving some
desirable common property (e.g. pairwise disjointness, partition of unity, etc.) of a
collection of functions, when the entire collection is extended simultaneously and
continuously from a closed subset A of a normal topological space X to the whole
space. The functions treated here are mainly real-valued, but in the last section, a
procedure introduced by Dugundji is used to show how to preserve the equicontinuity
of a collection of functions whose ranges lie in a locally convex metric linear
space.
