Analysis proof showing discontinuous funtion is integrable?

  • Thread starter mali0462
  • Start date
  • #1
3
0
analysis proof...showing discontinuous function is integrable?

Homework Statement


if a function f : [a,b] is Riemann integrable and g :[a,b] is obtained by altering values of f at finite number of points, prove that g is Riemann integrable and that
∫ f = ∫ g (f and g integrated from a to b)




Homework Equations





The Attempt at a Solution



g is bounded on [a,b] so for all E>0 let Q be a partition of [a, b] such that
PcQ

then L(P,f)<L(Q,g)<U(Q,g)<U(P,f) (inequalities should be less than or equal
to...how to type that?)

therefore U(Q,g)-L(Q,g)<E

therefore g is Riemann integrable on [a,b]
 
Last edited:

Answers and Replies

  • #2
StatusX
Homework Helper
2,564
1
What is P? How did you derive those inequalities?

The important point is that the points are only going to affect a finite number of grids in a given partition, and by arranging these grids to be small enough, you can make their effect negligible.
 
  • #3
3
0
P is my partition of f, (f given as integrable) and the inequalities are given in a theorem. I think I am trying to do what you said. Trying to set up Q, my partition of g, as constants plus or minus a delta term and then deriving my U(Q,f) and L(Q,f). Does that make sense?
 
  • #4
StatusX
Homework Helper
2,564
1
What theorem is specific enough that you can just write those inequalities down given the relation between f and g? And no, that didn't make sense (to me at least). Remember that there is no specific partition of f or g, you need to show that the result is the same over all partitions as their meshes go to zero.
 
Last edited:
  • #5
3
0
Alright. I see your point here, my inequality set up doesn't work yet. Thanks for your help thus far. I will go try something else. And I knew that I have to do this
"The important point is that the points are only going to affect a finite number of grids in a given partition, and by arranging these grids to be small enough, you can make their effect negligible." but any further hints on how to do that.
 
  • #6
StatusX
Homework Helper
2,564
1
Say the mesh (width of the largest grid) is e, and the places where g differs from f are x_1, x_2,..., x_n. Then what is the biggest the difference between the sums for f and g could be in terms of f(x_k), g(x_k), and e?
 
  • #7
matt grime
Science Advisor
Homework Helper
9,395
3
Why is g bounded?

HINT: it is easy, and equivalent, to consider only a function g that is zero except at a finite number of points.
 

Related Threads on Analysis proof showing discontinuous funtion is integrable?

Replies
1
Views
2K
  • Last Post
Replies
4
Views
1K
Replies
4
Views
6K
  • Last Post
Replies
3
Views
2K
Replies
7
Views
3K
Replies
4
Views
2K
  • Last Post
Replies
2
Views
5K
Replies
7
Views
2K
Replies
0
Views
783
Replies
4
Views
3K
Top