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BustedBreaks
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Let f:[0,1] be defined as f(x)= 0 for x rational, f(x)=x for x irrational
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!
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!