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## Main Question or Discussion Point

So I've got my exam of analysis tomorrow, but there's this piece of improper integrals I just don't get... (I'll paraphrase the definition we saw into english)

Now how can you see that if f is impr. int. over ]a,b[, that it is also impr. int. over ]a,c] and [c,b[ for a certain c in ]a,b[? It was noted in my papers as "evident", but I have no clue of how to prove it (and believe me, I've tried...). The main problem is (with the essence sketched) that with "x + y converges" given, that I have to proof x and y converge seperately. Obviously this is not true in all contextx, but apparently for integrals it is. I know it depends on the fact that in the definition you say the convergence to a limit should be independent of any rows c_n, d_n, but I don't know how to milk that...

(I see other courses sometimes define and improper integral over ]a,b[ as actually the sum of improper integrals over ]a,c] and [c,b[ and then surely it is evident, but as my course does not take that road, I'd like to be able to show it's equivalent...)

Thank you very much, I'm clueless.

*So if f is not necessarily Rieman-integrable over ]a,b[ (a and b can be (negative) infinity), but for all c,d with a<c<d<b [tex]\int_c^d f[/tex] exists and the limit of this integral for [tex]c \to a, d \to b[/tex] exists independent of how you approach a and b (relatively to each other), then we call f improperly integrable over ]a,b[.*Now how can you see that if f is impr. int. over ]a,b[, that it is also impr. int. over ]a,c] and [c,b[ for a certain c in ]a,b[? It was noted in my papers as "evident", but I have no clue of how to prove it (and believe me, I've tried...). The main problem is (with the essence sketched) that with "x + y converges" given, that I have to proof x and y converge seperately. Obviously this is not true in all contextx, but apparently for integrals it is. I know it depends on the fact that in the definition you say the convergence to a limit should be independent of any rows c_n, d_n, but I don't know how to milk that...

(I see other courses sometimes define and improper integral over ]a,b[ as actually the sum of improper integrals over ]a,c] and [c,b[ and then surely it is evident, but as my course does not take that road, I'd like to be able to show it's equivalent...)

Thank you very much, I'm clueless.