1. Suppose that a continuous function f : (a, b) -> R is differentiable except possibly at xo E (a, b), and that lime-+x0 f' (x) = L exists. Prove that f' (To) = L. Warning: It is NOT TRUE in general that lime-a limy +b f (x, y) = limy +blimx-+a f(x, y), even for beautiful functions f (x, y). Don't switch the order of nested limits without careful checking! 2. Suppose that f is continuous on a, b and differentiable on (a, b), but the derivative may be discontinuous. Prove that if f' (@) is monotone, then f'(x) is continuous on (a, b). 3. Show that if f' (x) exists and is continuous in some interval (c, d) containing x = a, and if f'(a) > 0, then there is some h > 0 such that f(x) is strictly increasing on the interval [a - h, a + h]