(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

f is a function defined in [0,1] by:

If there's some n (natural number) so that x = 1/n then f(x) = 1

If not then f(x) = 0.

Prove or disprove:

1) For every E>0 there's some P (P is a division of [0,1], for example

{0,0.2,0.6,1}) so that S(P) < E (Where S(P) is the upper some of f(x) with the division P)

2)f in integrable in [0,1] and the integral is 0.

2. Relevant equations

3. The attempt at a solution

1) True, because we can always choose A<E and: P={0,A,1/n,...,1/2,1} were 1/n is the smallest number of that type bigger than A. And then

S(P) =A < E

2)True, because if for every E>0 there's someP so that S(P) < E than the upper integral of f in [0,1] is 0. And since the lower integral is always bogger or equal to 0 (in this case) but smaller than the upper integral then the upper and lower integrals are both equal to 0 and so the integral exists and is 0.

Are thise right?

Thanks.

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# Homework Help: Integral question

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