Could someone give me an idea for a proof of this

  • #1
1,444
2
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.
 

Answers and Replies

  • #2
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,950
19
Can you prove it's integrable over [a, 1] for some positive a?
 
  • #3
NateTG
Science Advisor
Homework Helper
2,450
6
What definitions are you using? You should have no problem showing that the upper and lower sum converge.
 
  • #4
1,444
2
how

Hurkyl said:
Can you prove it's integrable over [a, 1] for some positive a?

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
HallsofIvy
Science Advisor
Homework Helper
41,847
966
stunner5000pt said:
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...?

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
1,444
2
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 ??
 

Related Threads on Could someone give me an idea for a proof of this

  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
881
Replies
7
Views
1K
Replies
0
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
4
Views
15K
Replies
2
Views
5K
Top