Hi everybody! I'm currently studying integrals, and I would like to clarify a few definitions, especially about the criterions of convergence/divergence of an integral. Basically if that's okay for you guys I'm gonna list and number a few statements and I'd like to know if they are true or not.(adsbygoogle = window.adsbygoogle || []).push({});

1. If adefiniteintegral∫_{a}^{b}f(x) dx hasbothits domain of integration and integrandbounded, but its set of locations where it is not continuous is anuncountableset, then f(x) is not integrable.

(or do we have to run other tests to find out if the function is integrable or not?)

2. If adefiniteintegral∫_{a}^{b}f(x) dx hasbothits domain of integration and integrandunbounded, then it is called animproper integral. While the term improper integral normally designates thelimitof an unbounded integral, this "ambiguity is resolved as both the proper and improper integral will coincide in value" (Wikipedia, improper integral).

3. Theimproper integralof a function f(x) exists, if there exists a functiong(x)so that ∫_{a}^{b}g(x) dxconvergesand|f(x)| ≤ g(x)∀ x ∈ [a,b). If one of those criterions is not met, then we cannot conclude anything about the existence of the improper integral of f(x) and we must run other tests. (Majorant criterion)

4. Theimproper integralof a function f(x)doesn't exist, if there exists a functiong(x)so that ∫_{a}^{b}g(x) dxdivergesandf(x) ≥ g(x)∀ x ∈ [a,b). If one of those criterions is not met, then we cannot conclude anything about the non-existence of the improper integral of f(x) and we must run other tests. (Minorant criterion)

I stop here, it would already mean a lot to me if those simple assumptions would become facts. :)

Thank you very much in advance, I appreciate your help.

Julien.

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# I General questions about integrals (definitions)

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