*8. Let I be continuous on a closed, bounded interval [a, b] and suppose that DRI(x) exists for all x E

(a, b).

(a) Show that if I

(b) < Yo < I(a), then Xo := sup{x E [a, b] : I(x) > yo}

satisfies I(xo) = Yo and DRI(xo) -::;. O.

(b) Prove that if I

(b) < I(a), then there are uncountably many points x that satisfy DRI(x) -::;. O.

(c) Prove that if DRI(x) > 0 for all but countably many points x E

(a, b), then I is increasing on [a, b].

(d) Prove that if DRI(x) ;::: 0 and g(x) = I(x) + x/n for some n E N, then DR9(X) > O.

(e) Prove that if DRI(x) ;::: 0 for all but count ably many points x E

(a, b), then I is increasing on [a, b]