Could someone give me an idea for a proof of this

1. Mar 30, 2004

stunner5000pt

Let f ( x ) = {0, if x = 1/n for some natural number n
or 1, otherwise

Note: Natural number would refer to the set of positive integers Z+ that is 1,2,3,...

Prove that this function is integrable on [0,1] and it's integral is 1

Certainly there are an infinite number of dicontinuities and nearly all of the function lies in the domain of [0,1]. But is the set of Inverse Naturals (1/n) (postive integers) bigger than the set of irrationals?

Someone recommended using the Robustness of the Reimann Integral

Let g and f be two functions defined on [a,b], and suppose
that the set of numbers in [a,b] at which the functions do not
take the same value (at which they "differ") is finite.

2. Mar 30, 2004

Hurkyl

Staff Emeritus
Can you prove it's integrable over [a, 1] for some positive a?

3. Mar 30, 2004

NateTG

What definitions are you using? You should have no problem showing that the upper and lower sum converge.

4. Mar 30, 2004

stunner5000pt

how

what do you mean? AS for the upper and lower sums - so then

U(f,P) = Sum i =1 to infinity 1 = infinity

and L (f,P) = 0

they dont seem to converge...?

5. Mar 30, 2004

HallsofIvy

Staff Emeritus
Oh, dear. I don't want to hurt your feelings but you are way, way off.
For one thing, your integral is from x= 0 to x= 1 so the sum is NOT from 1 to infinity. The sum in U(f,P) is over the intervals in the partion- for any finite partition, it is a finite sum. Also the summand is not 1:it is 1 times the length of the interval in the partition. U(f,P) is NOT infinity.

Also, L(f,P) is not 0. Since f(x) is 1 for all x except 1/2, 1/3, 1/4, etc. it clearly is equal to 1 for all x> 1/2: L(f,P) will be 1(1/2)+ something.

Oh, by the way:
"But is the set of Inverse Naturals (1/n) (postive integers) bigger than the set of irrationals?"

Not even close: there is an obvious one-to-one correspondence between the naturals and "inverse naturals" (n<-> 1/n) so the set {1/n} is countable.

6. Mar 30, 2004

stunner5000pt

so how can i prove taht the set of naturals is countable or finite, thus proving that the function has a finite number of discontinuities?

is there a theorem ??