a. Suppose that f: (0, 1) R is a non-negative continuous function. Show that (0, 1) exists

Question:

a. Suppose that f: (0, 1) → R is a non-negative continuous function. Show that ∫ (0, 1) exists if and only if lim Є→ ∫ c 1-c f exists.
b. Let An = [1 - 1/2n, 1 - 1/2n +1] Suppose that f: (0, 1) →R satisfies ∫Arf = (-1)n/n and f(x) = 0 for all x Є Un An. Show that ∫(0,1)f does not exist, but limЄ→∫(Є, 1 - Є)f = log 2.