3b) show an example for a function wich is bounded and exists in [a,b]

but its not integrabile.

prove that its not integrabile by definition

3c)calculate

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- September 2nd 2011, 04:45 AMtransgalacticintegrability question
3b) show an example for a function wich is bounded and exists in [a,b]

but its not integrabile.

prove that its not integrabile by definition

3c)calculate - September 2nd 2011, 06:16 AMHallsofIvyRe: integrability question
What have

**you**done on this? - September 2nd 2011, 06:38 AMtransgalacticRe: integrability question
i dont know how to solve th first part

but i know some laws for proving integrability

(*) for some dividing P

E is the set of uppers sums, T is the set of lower sums

if infE=supT then its defferentiable.

(*) S(p) is the sum of each supremum of the function in subsection mutiplied by the length of the subsection.

s(p) is the sum of each supremum of the function in subsection mutiplied by the length of the subsection.

in each subsection if inf{S(p)-s(p)}=0 then its defferentiable

or if

__ok i tried this one:__

f(x)=1 on x=0

f(x)=0 on other wise

for every diving of the section s(p)=0 so is supT=0 too

but S(p) is obviosly not sero so here is my example.

so but i dont know how to express S(p)

?

regarding the second one i thought of using by parts

but its not solving it - September 2nd 2011, 11:37 AMe^(i*pi)Re: integrability question
For the second part integration by parts is a good start (you'll then want to try partial fractions)

Now use partial fractions.

Hint: - September 2nd 2011, 12:04 PMtransgalacticRe: integrability question
- September 2nd 2011, 12:11 PMSironRe: integrability question
Use the hint of e^(ipi) which is very useful:

EDIT:

If you would use the traditional method then it has to be:

That means:

therefore and

And so:

- September 2nd 2011, 12:16 PMtransgalacticRe: integrability question
ok i could take care of this integral from here

regarding the first part:

i know some laws for proving integrability

(*) for some dividing P

E is the set of uppers sums, T is the set of lower sums

if infE=supT then its defferentiable.

(*) S(p) is the sum of each supremum of the function in subsection mutiplied by the length of the subsection.

s(p) is the sum of each supremum of the function in subsection mutiplied by the length of the subsection.

in each subsection if inf{S(p)-s(p)}=0 then its defferentiable

or if

ok i tried this one:

f(x)=1 on x=0

f(x)=0 on other wise

for every diving of the section s(p)=0 so is supT=0 too

but S(p) is obviosly not sero so here is my example.

so but i dont know how to express S(p)

? - September 2nd 2011, 02:29 PMmr fantasticRe: integrability question
At the level you appear to be studying at, you are most surely expected to know how to perform a partial fraction decomposition. You can at least work backwards from your answer and see if it works.

And please show all of your working in your first posting of future questions. We should not have to be asking for it. - September 2nd 2011, 10:05 PMtransgalacticRe: integrability question
my last post states that i have solved the integral and moved to the next one.

regarding the first part:

i know some laws for proving integrability

(*) for some dividing P

E is the set of uppers sums, T is the set of lower sums

if infE=supT then its integrible.

(*) S(p) is the sum of each supremum of the function in subsection mutiplied by the length of the subsection.

s(p) is the sum of each supremum of the function in subsection mutiplied by the length of the subsection.

in each subsection if inf{S(p)-s(p)}=0 then its defferentiable

or if

ok i tried this one:

f(x)=1 on x=0

f(x)=0 on other wise

for every diving of the section s(p)=0 so is supT=0 too

but S(p) is obviosly not sero so here is my example.

so but i dont know how to express S(p)

? - September 3rd 2011, 01:14 PMemakarovRe: integrability question
It is known that a function is Riemann integrable if and only if it is bounded and continuous almost everywhere, i.e., the set of its points of discontinuity has measure zero. A set has measure zero if it can be covered by intervals whose combined length can be made arbitrarily small; however, it is not necessary to understand this precisely for this problem. Every finite set and even every countable infinite set has measure zero, so the function you tried: f(x)=1 on x=0 f(x)=0 otherwise, is integrable (it has a single point of discontinuity). You need a lot more points of discontinuity. Consider Dirichlet function, which is nowhere continuous. It is easy to show that the upper sum, say, on [0, 1] is always 1 and the lower sum is 0.

Quote:

(*) for some dividing P

E is the set of uppers sums, T is the set of lower sums

if infE=supT then its defferentiable.

- September 3rd 2011, 01:50 PMtransgalacticRe: integrability question
dirichlet functio is defined

D(x) =1 for rational x

D(x) =0 for irrational x

so supT=0

so infE=1

supT is not equal infE so its not integrible

correct? - September 3rd 2011, 02:04 PMemakarovRe: integrability question
To say whether this is correct, I need to know what are , , T and E, as well as why all and all .