Utilizing Continuous Selection in the Definition of Continuity

The main result of this thesis deals with continuous functions on metric spaces. Specifically, we show that given a continuous function f : M1 → M2 from a metric space (M1, d1) to a metric space (M2, d 2), there is a function delta : M1 x R+→R+ such that delta is continuous and, if ∀x ∈ M1 ∀epsilon > 0 ∀y ∈ M1 [d1(x, y) < delta(x, epsilon) ⇒ d2 (f (x), f (y)) < epsilon]. The above result was established by Enayat in [Ena00] using the machinery of partitions of unity. Our expository account of Enayat's paper contains a substantial body of results in general topology, including a thorough discussion of paracompact spaces and partitions of unity.