MATH 350, Homework 10, problems 1-2.
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Let f and g be real-valued, bounded
functions defined over some domain D.
- Prove that if ∀ x ∈ D f(x) ≤ g(x) then
sup f(x) ≤ sup g(x) and inf f(x) ≤ inf g(x).
- If ∀ x ∈ D ∀ y ∈ D f(x) ≤ g(y) then
sup f(x) ≤ inf g(x).
- 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.
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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).
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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).
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.