Spaces of Continuous Functions

In this chapter we shall apply the theory we developed in the previous chapter to spaces where the elements are continuous functions. We shall study completeness and compactness of such spaces and take a look at some applications. 2.1 Modes of continuity If (X, d X) and (Y, d Y) are two metric spaces, the function f : X → Y is continuous at a point a if for each > 0 there is a δ > 0 such that d Y (f (x), f (a)) < whenever d X (x, a) < δ. If f is also continuous at another point b, we may need a different δ to match the same. A question that often comes up is when we can use the same δ for all points x in the space X. The function is then said to be uniformly continuous in X. Here is the precise definition: Definition 2.1.1 Let f : X → Y be a function between two metric spaces. We say that f is uniformly continuous if for each > 0 there is a δ > 0 such that d Y (f (x), f (y)) < for all points x, y ∈ X such that d X (x, y) < δ. A function which is continuous at all points in X, but not uniformly continuous, is often called pointwise continuous when we want to emphasize the distinction. Example 1 The function f : R → R defined by f (x) = x 2 is pointwise continuous, but not uniformly continuous. The reason is that the curve becomes steeper and steeper as |x| goes to infinity, and that we hence need increasingly smaller δ's to match the same (make a sketch!) See Exercise 1 for a more detailed discussion.