Let f:[0,1] be defined as f(x)= 0 for x rational, f(x)=x for x irrational(adsbygoogle = window.adsbygoogle || []).push({});

Show f is not integrable

m=inf(f(x) on [Xi-1, Xi])

M=sup(f(x) on [Xi-1, Xi])

Okay so my argument goes like this:

I need to show that the Upper integral of f does not equal the lower integral of f

Because rationals and irrationals are dense in R for any interval [a,b] of f,

m=0

M=x

therefore, for the partition X0=0, X1=1 the Lower Darboux sum = 0 and the Upper Darboux sum = x

therefore the sup{Lower Darboux sum}=0

and the inf{Upper Darboux sum} is not equal to 0, (it is the smallest irrational number greater than 0 technically right? although there is no smallest irrational greater than 0, right?)

therefore the Upper integral of f does not equal the lower integral of f

and therefore f is not integrable.

I think this is right, but I just want to make sure I didn't miss anything.

Thanks!

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# Homework Help: Prove not integrable. Is this correct?

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