# Thread: ssplitted function integrability prove

1. ## ssplitted function integrability prove

3)b)
there is f(x) a continues and positive in [0,1]

g(x)=f(x) for rational x
g(x)=-f(x) for irational x

prove that g(x) is not integrabile in [0,1]
?

how i tried to solve it:
suppose that this not true and it integrabile
so so every dividing p
inf{S(p)-s(p)}=0
$\displaystyle S(p)=\sum_{i=1}^{n}M_{i}(x_{i}-x_{i-1)}$
$\displaystyle s(p)=\sum_{i=1}^{n}m_{i}(x_{i}-x_{i-1)}$
or i need to prove that infE=supT
now i need to get expresstion and prove that the equations above cannot exist.

what to do here in general
?

2. ## Re: ssplitted function integrability prove

One simple way: the set of discontinuity points of $\displaystyle f$ is $\displaystyle [0,1]$ . Taking into account that $\displaystyle [0,1]$ has not measure zero, $\displaystyle f$ is no Riemann integrable (as a consequence of a well known theorem).

3. ## Re: ssplitted function integrability prove

ok i understand the theory that because of endless points of discontinuety its not differentiable.

how in general could i show this using the above formulas i wrote
?