- #1

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1. If a

**definite**

**integral**∫

_{a}

^{b}f(x) dx has

**both**its domain of integration and integrand

**bounded**, but its set of locations where it is not continuous is an

**uncountable**set, 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 a

**definite**

**integral**∫

_{a}

^{b}f(x) dx has

**both**its domain of integration and integrand

**unbounded**, then it is called an

**improper integral**. While the term improper integral normally designates the

**limit**of an unbounded integral, this "ambiguity is resolved as both the proper and improper integral will coincide in value" (Wikipedia, improper integral).

3. The

**improper integral**of a function f(x) exists, if there exists a function

**g(x)**so that ∫

_{a}

^{b}g(x) dx

**converges**and

**|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. The

**improper integral**of a function f(x)

**doesn't exist**, if there exists a function

**g(x)**so that ∫

_{a}

^{b}g(x) dx

**diverges**and

**f(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.