# Analysis proof showing discontinuous funtion is integrable?

• mali0462

#### mali0462

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)

## 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]

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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.

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?

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.

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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.

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?

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.