1873_solutions

# N continue as above of course we dont have a guarantee

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Unformatted text preview: nd that Þa sn  Þa f . b b b. Suppose that u is a decreasing differentiable function on an interval a, b and that the derivative u of u is integrable on a, b . Apply the form of the monotone version of the change of variable theorem proved above to the function v defined by the equation v t  u t for b t a to show that the equation Þa f u t b u t dt  Þu a ub f x dx holds for every function f integrable on the interval u b , u a . We define g t  f t for all t a, b . Then g is integrable on Þ ua ub g Þu b ua Þu a ua, ub and we have ub f f From the monotone version of the theorem proved earlier we also see that Þ ua g Þv a g  ub Þa g ut vb Þa g v t b b v t dt u t dt  Þa f u t b u t dt. We conclude that Þa f u t b u t dt  Þu a ub f To reach some additional exercises that invite you to develop some important inequalities, click on the icon . Exercises that Yield Another Version of Darboux’s Theorem The exercises in this section depend upon the material on Darboux’s theorem. Their purpose is to show that if f is Riemann-Stieltjes integrable with respect to an increasing function  and if  is continuous at every number at which  jumps then Darboux’s theorem can be stated using the ordinary mesh P of a partition P instead of the -mesh. 1. Prove the following special case of Darboux’s theorem that applies when  is continuous: Suppose that f is Riemann-Stieltjes integrable with respect to an increasing continuous function  on an interval a, b . Then for every number  0 there exists a number   0 such that for every partition P  x 0 , x 1 , , x n 317 of a, b satisfying the inequality P   and every choice of numbers t j n f tj  xj  xj Þ a fd b 1 j1 x j 1 , x j for each j we have . Since the function  is continuous on the interval a, b , it is uniformly continuous there. Suppose that  0. Using Darboux’s theorem, choose a number  1  0 such that the inequality n f tj  xj  xj Þ a fd b 1 j1  will hold whenever the partition P  x 0 , x 1 , , x n satisfies the condition , P   1 . Using the fact that  is uniformly continuous on the interval a, b , choose   0 such that the inequality , P   1 will hold whenever P  . 2. Suppose that  is an increasing function that varies discretely on an interval a, b . Suppose that f is a bounded function on a, b and that f is continuous at every number at which the function  is discontinuous. Prove that for every number  0 there exists a number   0 such that for every partition P  x 0 , x 1 , , x n x j 1 , x j for each j, if we of a, b satisfying the inequality P   and for every choice of numbers t j define g to be the step function that takes the value f x j at each number x j and that takes the constant value f t j in each interval x j 1 , x j then we have Þ a gd Þ a fd b b . Suppose that  0. We write the set of discontinuities of  in the interval a, b as y 1 , y 2 , y 3 ,  . Choose a number k such that the inequality |f x | k holds for every num...
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## This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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