I just finished taking my intro Real Analysis class and our final unit was on sequences of functions. We talked about how a function can be pointwise convergent (ex: fn(x) = x^n on [0,1] converges PW to f(x) = {1 if x=1, 0 otherwise}) and then how this doesn’t really tell us much in terms of function properties (continuity, differentiability, Riemann Integrability) even while knowing each fn is continuous, differentiable, etc. Because of this, we use Uniform Convergence instead as this lets us reason about continuity and RI. My question is why do we even care about Pointwise convergence then if it seemingly gives us no real information to work with? What can we do with JUST the Pointwise convergence of a sequence of functions that makes it worthwhile?
While writing, I realized it may be useful for showing something isn’t uniform convergence (because uniform implies pointwise so contrapositive). However this use case seems isolated.
When you will learn about measure theory and Lebesgue integration, you will appreciate pointwise convergence much more. The theorems about the Lebesgue integral require only pointwise convergence + some other condition. Actually, this is one of the main advantages of Lebesgue integration over Riemann integration, as there are lots of sequences which only converge pointwise, but not uniformly. (and even if they do converge in both senses, checking uniform convergence is much more difficult)
Besides, pointwise convergence not only helps to prove a sequence is not uniformly convergent. When you want to prove it is uniformly convergent, it's much easier to do if you know what the limit function is. For this you first have to check pointwise convergence.