1) I assume your domain is notice that to evaluate we need three cases:
-) If then
-) If then
-) If then and for big enough.
2) Notice that are Riemann integrable.
Suppose that and converges uniformly to f [/tex], by a theorem that I know, and provided that , X being the domain set.
Now, if , this may not be true.
Counterexample:
Define
And define
And uniformly, but is integrable for all n while f is not.
Question:
Q1: Why is uniformly? The book just claim it is, but I'm trying to prove it using the defintion of such that
Q2: I understand why f is not integrable since it goes to infinity, but are integrable? Is it because is finite?
Thank you!