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Exercise 3.6.3: Finish the proof of Corollary 3.6.3. Corollary 3.6.3. If I C R is an interval and f: I R is monotone and not constant, then f(I) is an interval if and only if f is continuous. Assuming f is not constant is to avoid the technicality that f (I) is a single point in that case; f (I) is a single point if and only if f is constant. A constant function is continuous. Proof. If f is continuous then f (I) being an interval is a consequence of intermediate value theorem. See also Exercise 3.3.7. Let us prove the reverse direction by contrapositive. Suppose f is not continuous at c EI, and that C is not an endpoint of I. Without loss of generality suppose f is increasing. Let a := lim f(x) = sup {f (x) : X EI, X < c}, b := lim f '(x) = inf{f(x) : X € I, X > c}. . x-sct As C is a discontinuity, a < b. If X < c, then f(x) a, and if X > c, then f ( x) > b. Therefore any point in (a,b) \ {f(c)} is not in f (I). . However there exists X1 € S, X1 < C so f (x1) a, and there exists X2 ES, X2 > c so f(x2) > b. Both f(x1) and f(x2) are in f (I), , and so f(I) is not an interval. See Figure 3.6. When C € I is an endpoint, the proof is similar and is left as an exercise. Exercise 3.6.8: Suppose f: I J is a continuous, bijective (one-to-one and onto) function for two intervals I and J. Show that f is strictly monotone. Exercise 3.6.9: Consider a monotone function f: I R on an interval I. Prove that there exists a function g: I R such that lim g(x) = g(c) for all c € I, except the smaller (left) endpoint of I, and such that g (x) = f(x) for all but countably many X. Exercise 4.4.3: Let n € N be even. Prove that every X > 0 has a unique negative nth root. That is, there exists a negative number y such that yn = X. Compute the derivative of the function g(x) :=y. Exercise 4.4.6: Let f(x) == X + 2x2 sin (1/x) for X # 0 and f(0) = 0. Show that f is differentiable at all X, that f'(0) > 0, but that f is not invertible on any interval containing the origin. Exercise 4.4.7: a) Let f: R R be a continuously differentiable function and k > 0 be a number such that fl (x) > k for all X € R. Show f is one-to-one and onto, and has a continuously differentiable inverse f-1: R R. b) Find an example f : R R where f'(x) > 0 for all X, but f is not onto. Exercise 5.2.2: Let f and g be in Re/a,b]. Prove that f + g is in 1. b (f (x) +8(x)) dx= 1. b f (x) dx+ 1. b g(x) dx. a a Hint: Use Proposition 5.1.7 to find a single partition P such that U(P,f) - L(P,f) < 8/2 and U (P,8) - L(P,8) < 8/2. Proposition 5.1.7. Let f: [a,b] R be a bounded function, and let P be a partition of [a,b]. Let P be a refinement of P. Then L(P,f)<<(P) and U(P.A) Exercise 5.4.4: Use the geometric sum formula to show (fort+-1) 1 = 1+t 1 1+t Using this fact show = for all X € (-1,1] - (note that X = 1 is included). Finally, find the limit of the alternating harmonic series 8 ( - 1) n+1 = 1-1/2+1/3-1/4+ - n n=1 Exercise 5.4.5: Show n ex = n-100 lim Hint: Take the logarithm. Note: The expression arises in compound interest calculations. It is the amount of money in a bank account after 1 year if 1 dollar was deposited initially at interest X and the interest was compounded n times during the year. Therefore ex is the result of continuous compounding. Exercise 5.4.10: Show that E (x) = ex is the unique continuous function such that E(x+y) = E (x) E(y) and E (1) = e. Similarly prove that L (x) = In(x) is the unique continuous function defined on positive X such that L(xy) = L (x) + L (y) and L(e) = 1.

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