Reference for theory of non-uniformly convergent sequences of functions

I have been thinking about some conjectures of mine regarding non-uniformly convergent sequences of functions. I'm happy to discuss my work so far, if anyone wants to know, but for brevity let me just say that I am particularly interested in finding a reference which discusses some theory of non-uniformly convergent sequences of functions, specifically for functions from R2->R. Does anyone know of a good place to look for this?.

I have been trying to find a reference today for this subject and could not, is that because nobody thinks it's very interesting? I tried google, wiki, googlebooks, ect... but there are SO many ref's for uniform convergence that come up, it's hard to find what I'm looking for.

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I don't follow. Do you want sequences that converge, just not uniformly? In what topology? There are plenty of other possible topologies to consider, and that's what functional analysis is for. –  Qiaochu Yuan Aug 24 '10 at 22:51
I want sequences which converge, and the convergence should be non-uniform. I'm just using the standard euclidean norm. The neat thing about these sequences is that they can converge to different functions in different regions of the phase plane. e.g. f(n) = n^2x/n^2(x+1) goes in the limit to 1/x unless x=0, then its just the zero function. stuff.mit.edu/afs/athena/course/18/18.06-esg/18.034/notes/… –  Matt Calhoun Aug 24 '10 at 23:09
So, pointwise convergence? If you use that keyword you should find plenty of references. (You should specify this. There are lots and lots of ways for sequences of functions to converge, and pointwise convergence is one of the most useless.) –  Qiaochu Yuan Aug 24 '10 at 23:31
As for your conjectures, you can post them as questions if you want. The answers may be well-known. –  Qiaochu Yuan Aug 24 '10 at 23:34
@Matt, if a sequence converges pointwise, it converges to one function—although the limit function may not be writable with one equation. –  Mariano Suárez-Alvarez Aug 24 '10 at 23:53