To prove Theorem 1, (that If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + C) let F and G be any two antiderivatives of f on I and let H = G - F. (a) If x1 and x2 are any two numbers in I with x1 less than x2, apply the Mean Value Theorem on the interval [x1, x2] to show that H(x1) = H(x2). Why does this show that H is a constant function? (b) Deduce Theorem 1 from the result of part (a).

25. Let, where and are non empty, and let and be subsets of . Prove that. Prove that. Prove that. Prove that if.

If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xy

1.49. (!) Let f and g be functions from R to R. For the sum and product of f and g (see Definition 1.25), determine which statements below are true. If true, provide a proof; if false, provide a counterexample. a) If f and g are bounded, then f + g is bounded. b) If f and g are bounded, then fg is bounded. c) If f + g is bounded, then f and g are bounded. d) If fg is bounded, then f and g are bounded. e) If both f + g and fg are bounded, then f and g are bounded. Y

4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove that f is constant.

Find continuous functions f1, f2, f3, f4 defined on the open unit interval I = (0, 1) (i.e. functions fi : (0, 1) → R) such thatˆ f1(I) = (L1, M1) ˆ f2(I) = (L2, M2] ˆ f3(I) = [L3, M3) ˆ f4(I) = [L4, M4]for some real numbers Li, Mi.

Q1 Theorem 18.1 states that "Let f be a continuous real-valued function on a closed interval [a, b]. Then f is a bounded function.". Read the proof of Theorem 18.1 with [a, b] replaced by (a, b). Where does it break down? Explain.