**Problem Statement:** For every and define . Find the function that converges to pointwise. Is the converges uniform?

**Proof:** Fix . Then and so converges to pointwise.

*Claim:* This convergence is not uniform.

If the convergence is uniform then it follows that for every we may find an such that for every and for every . Let and . Since this implies that . This is clearly a contradiction and so we have found a counterexample such that the sequence breaks uniform continuity. Thus, converges to pointwise but not uniformly.

**Reflection:** This one tripped me up a bit. I was able to get the piecewise convergence but I could not decide whether it converged uniformly or not. My intuition told me that it didn’t, but I couldn’t think of a counter example like above where it broke uniform convergence. While trying to find a counter example I saw that as got closer to the needed to make converge got larger and larger. This made me think that we couldn’t find a universal that would work for all . After seeing the above counter example it makes a lot more sense. I think what I need to keep in mind when trying to find a counter example for these kind of things is that I don’t necessarily want to find a specific value where it breaks, but that I should tailor my value to what I know, i.e. the .

The way that I identified immediately, that the convergence would not be uniform, is that I thought about extending the function to [0,1].

But when x=0, fn(x) = 1. And so you would have a sequence of continuous functions converging to something discontinuous on a compact space! So the convergence couldn’t have been uniform.

Also, as you said, you should always look for a breaking point that depends on n. No specific point could work due to pointwise convergence.

Alternatively, using a continuity argument, you could say that fn(0)=1 and fn(1) is less than or equal to 1/2 for any n. So by the intermediate value theorem, there exists x in (0,1) s.t. fn(x)=.75. And since such an x exists for any n, convergence is not uniform. (The advantage of this is that you don’t even need to find any values of x)