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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 RiemannStieltjes 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
RiemannStieltjes 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 j1 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 j1 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.
 Fall '08
 STAFF
 Math, Calculus

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