By Marina Cohen

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Is it differentiable? 1 (Another FTC for the Cauchy Integral). If f is a continuous function on the interval [at b], and we define a function F Oli [a, b] by F(x) = C J~'C f(t) dt, then 1. F is differentiable on [a. b], 2. F' = f on [a, b], and 3. F is absolutely continuous on [a, b]. Proof. We have shown that the Cauchy integral of a continuous function is well defined, so F makes sense. To show that F is differentiable on [a, b], and that the derivative of F, F', is in fact equal to f, entails estimating the familiar expression F(x + '~: - F(x) _ f(x) 38 A Garden of Integrals where h.

Washington: Mathematical Association of America, 1994. 3. Courant, Richard, and Fritz John. Introduction to Calculus and Analysis. Vol. 1. New York: Wiley Interscience, 1965. 4. Young, Robert M. Excursions ill Calculus' An Interplay of the Continuous and the Discrete. Dolciani Mathematical Expositions. No. 13. Washington: MathematIcal Association of America, 1992. CHAPTER 3 The Ruemawon ~ntegra~ Reason with a capital R = Sweet Reason, the newest and rarest thing in human life, the most delicate child of human history.

The sets = = 1-1 ([Yk-l. Yk» = {X E [a, b] I Yk-l ~ I(x) < Yk} are disjoint with union [a. b]. Disregarding the empty sets (relabelling if necessary), pick a tag (point Ck) in each nonempty set, and fonn the sum (motivated by areas of rectangles as the height times the length of the base) as follows (see Figure 21): I(cl)-{length of ,-1 ([Yo, Yl)}+"'+ f(clI)·{length of ,-1 ([Yn-li Yn»)}. We then have LYk-l . {length of f- 1 ([Yk-l. Yk»} < L ,-I ,-I I(ck) . {length of < LYk . {length of (fYk-l, Yk»)} ([Yk-l.

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