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tulip007

Member

Registered: 2008-04-30

Posts: 10

Riemann integration

Define q(x) = 1 if x ε Q and g(x) = 0 otherwise.  Prove that q is not Riemann integrable on {0,1) by showing that for all partitions P
U (q, P) – L(q,P) > 1

I am really lost on this question, any help would be greatly appreciated!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Riemann integration

Are you certain that's the way the question is stated?  First off, when talking about Riemann integration, it is typical to integrate over compact sets, namely [0, 1] instead of (0, 1).  Second, to prove something is not Riemann integrable, it is only necessary to prove that:

L(q, P) ≠ U(q, P)

In fact, in this case, U(q, P) - L(q, P) = 1, so it makes no sense for you to be required to show that it is greater than 1.

That being said... you need to go by definition.  Pick an arbitrary partition {x_0, ..., x_n} such that:

Now what can be said about each of these intervals, [x_i, x_{i+1}]?  What value will U(q, P) and L(q, P) take on each?  Remember the density of the rationals and irrationals in R.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."