 # 1. (Bounded Monotone). Let f(x) be a bounded monotone increasing fu...

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1. (Bounded Monotone). Let f(x) be a bounded monotone increasing function on an interval (a,b). In this problem we prove that lim f(x) exists. (A similar argument could show lim f(x) also exists.) b (Note: f fxx is not necessarily defined or continuous at x = a, b. Recall that bounded means there exists an M so that f(x) < M for all x. Monotone increasing means that x1 < x2 a) Let A = {f(x) | x € (a,6)}. Justify why this has a greatest lower bound Y. b) Explain why for any € > 0 there must be an xo € (a,b) with f (xo) < Y + E. Use xo to find a 8 so that 0 - rl < € and complete the proof. c) Explain in words why this means that for bounded monotone functions on (a,6), at each point where c € (a,b) both lim f (x) and lim f(x) will exist. (Actually it can be shown that monotone functions can only have a countable number of discontinuities. Since countable sets have measure 0 this means monotone functions will be continuous almost everywhere. Very nicely behaved!) 2. (Total Separation). Let f(x) be continuous on [a, b]. Suppose there is a value L such that f (x) # L for any x € [a,b]. Complete the following steps to prove that there is an € > 0 such that f(x) - L > E for all x € [a, b]. a) Justify why either L > M or L < m where M and m is the max and min of f(x) on [a,b]. b) Use L and M to construct an E and show that f (x) - E for all x € [a, b] (and similarly for L and m). c) Show that the theorem fails if we replace [a, b] with the open interval (a,b). 3. (An Increasing derivative). Let f be differentiable on an interval I and f' (x) is increasing on I. a) Let T(x) be the tangent line at a point c in the interval. Write down the equation for T(x). b) Show for any x € I with x > c that f(x) > T(x). (Hint: consider the auriliary function h(x) = f(x) - T(x). Apply the MVT f on [c, x] and use what you know about f.) c) Show that the same holds true for x € I with x < c. That is f(x) > T(x). Thus when f'(x) is increasing the tangent line always lies below the function. 4. (Lipschitz again). Suppose f satisfies a Lipschitz condition of the form f(x) - f(y) < K|x - y on the interval [a,b]. We prove that f is integrable on [a,b]. a) Prove that f must be bounded on [a,b]. (Hint: Use the condition to show that f(a) - K(b - a) < f(x) < f (a) + K(b-a).) - b) Use the Lipschitz condition on the each subinterval xi] to show that f (xi) + K Ari is an upper bound for f on and f(xi) - Kxi is a lower bound for f on [xi-1,xi]. Conclude that - € c) Now given an € > 0 choose a partition Q of [a, b] so that each Axi Write out U ((Q) - L(Q) and, using part b, show that this is less than €. d) What theorem allows you to conclude that f is integrable on [a.b)? 5. a) (Reciprocal Integrability). Let f be integrable on [a,b]. Give an example to illustrate that in 1 general may not be integrable on [a,b]. f (x) b) Now assume f has a positive lower bound (i.e. f(x) > K > O for r € [a,b]). Using familiar notation let (i) M{ = sup {f(x) € [xi-1,xi]} - and = = x € xi and 1/f - Show that 0 < < 0 1 1 this implies 1 1 < c) Use this inequality to show that - 1 - 1 d) Use the inequality in part to show that if f is integrable then is also integrable. c f(x)

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