In this post we will be looking at uniform continuity.

To investigate the concept, let’s first look at what we already know. “Uniform” has shown up previously in the definition of uniform continuity. Comparing it to plain continuity will be insightful. Notice, then, how pointwise continuous functions have the sole requirement that we can meet any ε-challenge with a δ-neighborhood around the point in question, for which all values in the δ-neighborhood will map to be inside of the ε-neighborhood under the continuous function.

However, uniformly continuous functions have a slightly different definition. Instead of our δ possibly being dependent on the point in our domain, it is instead only dependent on ε and any other constant values. Therefore, if we are given an ε-challenge for a uniformly continuous function, it can be met by the same δ value at any point on the domain.

Now, let’s consider what is different with sequences of functions. Back to sequences, we’ve traded our δ in for N, but the concept is still the same. The definition of pointwise convergence is

Definition 6.2.1. For each n ∈ , let be a function defined on a set A ⊆ R. The sequence () of functions converges pointwise on A to a function f if, for all x ∈ A, the sequence of real numbers converges to f(x).

This definition from the book is slightly abridged but it does clearly show the important concept: that value at which we will take our sequence limit at is fixed first. As in, we choose x and then, at that x value, we take the limit.

The definition of uniform convergence is:

Definition 6.2.3 (Uniform Convergence). Let (fn) be a sequence of functions defined on a set A ⊆ R. Then, () converges uniformly on A to a limit function f defined on A if, for every ε > 0, there exists an N ∈ such that | − f(x)| < ε whenever n ≥ N and x ∈ A.

has a similar hallmark of the uniformly continuous definition: the point x is not introduced in the definition (besides as a dummy variable) until after N is produced. Therefore, to show uniform convergence, N must be chosen more carefully as to not depend on x.