# Thread: iintegrability question of irrational

1. ## iintegrability question of irrational

3)b)
there is a continues and positive f in [0,1]
prove that g(x) is not integrabile in [0,1]
g(x)=f(x) for rational x
g(x)=-f(x) for irational x
$\displaystyle S(p)=\sum_{i=1}^{n}M_{i}(x_{i}-x_{i-1})$

why its not integrible?
because f the endless points of disconinuety?

my prof showed a function
g(x)=1 for rational x
g(x)=1/x for irational x
and he said that it is integrible

i dont know why the original is not integrible
?

2. ## Re: iintegrability question of irrational

Regardless of the partition, the upper sums will be...

3. ## Re: iintegrability question of irrational

the upper sum will be
$\displaystyle S(p)=\sum_{i=1}^{n}Sup(g([0,1]))(x_{i}-x_{i-1})=\sum_{i=1}^{n}f(x)(x_{i}-x_{i-1})$
the lower sum
will be
$\displaystyle S(p)=\sum_{i=1}^{n}inf(g([0,1]))(x_{i}-x_{i-1})=\sum_{i=1}^{n}(-f(x))(x_{i}-x_{i-1})$

i dont know if the supremum is actually is f(x) because its not an actual number.
same thing for the infinum
i just guessed because this is the only thing we've got.

but in order to prove that its not integrabile
their subtraction shoudnld be lowe then epsilon
dont know how to show that

??

4. ## Re: iintegrability question of irrational

I should have hinted at the lower sum. On any interval, there will be a rational number, so the lower sums will be 1.
The uppers will not be one!

5. ## Re: iintegrability question of irrational

how you got to the conclution that f(x)=1
?

oohh i know that
my prof sayed that we are working only on darbu integral

what you are saying is lebeg integral
and its beyong the scope of my course

6. ## Re: iintegrability question of irrational

Originally Posted by transgalactic
3)b)
there is a continues and positive f in [0,1]
prove that g(x) is not integrable in [0,1]
g(x)=f(x) for rational x
g(x)=-f(x) for irational x
why its not integrible?
because f the endless points of disconinuety?
Assuming that you mean Riemann Integration.
Is it possible that $\displaystyle g$ is continuous at any point in $\displaystyle [0,1]~?$
Do you have a theorem on the cardinality of the set of discontinuities of an integrable function?

7. ## Re: iintegrability question of irrational

i just remmember the words of my prof that said that in our couse dereclet function is not integrible.
but in other courses it is integrible and the integral is 1

so when TheChaz said that the result is 1
i got remmemebered the above remark.