MATH 350, Homework 10, problems 1-2.

  1. Let f and g be real-valued, bounded functions defined over some domain D.

    1. Prove that if ∀ x ∈ D  f(x) ≤ g(x) then sup f(x) ≤ sup g(x) and inf f(x) ≤ inf g(x).
    2. If  ∀ x ∈ D  ∀ y ∈ D  f(x) ≤ g(y) then sup f(x) ≤ inf g(x).

    3. In the previous problem, sup f(x) = inf g(x) if and only if ∀ ε >0  ∃ xε, yε ∈ D such that g(yε) − f(xε) < ε.
    4. sup ( f(x) + g(x) ) ≤ sup f(x) + sup g(x);  inf ( f(x) + g(x) ) ≥ inf f(x) + inf g(x). (subadditivity and superadditivity).
    5. If k>0 then sup k f(x) = k sup f(x) and inf k f(x) = k inf f(x). If k<0 then sup k f(x) = k inf f(x) and inf k f(x) = k sup f(x).

  2. Let f(x)=x2, and [a,b]=[0,1]. Compute the n-th upper and lower Darboux sums on the uniform partition P_n={0,1⁄n,2⁄n,...,1}. Show directly that f is Darboux integrable on [0,1] and find the value of the Darboux integral.