Let f be continuous on a closed, bounded interval [a, b] and suppose that DRf{x) exists for all x ∈ (a, b).
a) Show that if f(b) < y0 < f(a), then
x0 := sup{x ∈ [a, b] : f(x) > y0}
satisfies f(x0) = y0 and DRf(x0) < 0.
b) Prove that if f(b) < f(a), then there are uncountably many points x which satisfy DRf(x) < 0.
c) Prove that if DRf(x) > 0 for all but countably many points x ∈ (a, b), then f is increasing on [a, b].
d) Prove that if DRf(x) > 0 and g(x) = f(x) + x/n for some n ∈ N, then DRg(x) > 0.
e) Prove that if DRf(x) > 0 for all but countably many points x ∈ (a, b), then f is increasing on [a, b].