1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Riemann integration in R^n

  1. Feb 24, 2010 #1
    1. The problem statement, all variables and given/known data
    a) Let A in R^n be compact, and let f: A -> R be continuous and also non-negative. Show that if there exists some a in A with f(a) > 0, then [itex]\int_{A}f > 0[/itex]
    b) Now let A in R^n be a closed rectangle, and suppose f: A -> R be bounded and integrable. Show that if f(x) > 0 for all x in A, then [itex]\int_{A}f > 0.[/itex]

    2. Relevant theorems
    A function is Riemann integrable iff its set of discontinuities has measure zero.

    3. The attempt at a solution
    a) For this, all we need is the fact that continuous functions are sign preserving right (of course f is integrable)? Specifically we can find a [itex]\delta > 0[/itex] such that for all x in the ball of radius [itex]\delta,[/itex] [itex]|f(x) - f(a)| < f(a) \Rightarrow f(x) > 0.[/itex] Then we can choose our partition so that some subrectangle, say S_i is contained in the ball. Hence the supremum (or infimum, I don't think it matters) of f over this subrectangle is actually equal to f(y) for some y in S_i due to compactness, so the upper sum (or lower sum) is > 0 (> 0 contribution from f over S_i, f is nonnegative elsewhere), so we are done. Does this work?

    b) I'm less sure about this one, but I think it follows from a) since we can find infinitely many points on A at which f is continuous (due to the relevant theorem), and we really only need one point. Again sups and infs are achieved due to compactness which follows from Heine-Borel.

  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted