1873_solutions

Now fill in the details why f must be strictly

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: t f is a continuous step function on an interval a, b and choose a partition P  x 0 , x 1 , , x n of a, b within which f steps. If c is the constant value of f on the interval x 0 , x 1 then, since f is continuous at x 0 and at x 1 we have f x 0  f x 1  c. Therefore, since f is continuous at x 1 , the number c must also be the constant value of f on x 1 , x 2 . Continuing in this way we see that f has the constant value c throughout the interval a, b . Exercises on Integration of Step Functions 1. Given that f is the function defined by the equation 0 1 fx  if x  1 if 1 x 2 1 if 2  x  4 2 if 4 x5 3 if 5 x7 0 if x 7 whose graph appears iin the figure 263 3 2 1 0 -2 2 4 6 8 10 -1 evaluate Þ Ý Ý f. Solution: We sum the function f over the partition P of the interval Ý 2, 1, 2, 4, 5, 7, 10 2, 10 . Þ0 f  0 1 2 1 2 1 2 2 5 14 4 3 7 5  0 10 7  7. 2. Prove that if f is a step function then so is the function |f | and we have Ý Ý Þ Ýf Þ Ý |f |. Choose an interval a, b outside of which the function f is zero. We deduce from Theorem 11.3.8 that Ý Þ Ýf  Þa f Ý Þ a |f |  Þ Ý |f |. b b 3. Given that f is a function defined on R and that the set of numbers x for which f x must be a step function and why 0 is finite, explain why f Ý Þ Ý f  0. We define a and b to be the smallest and largest members, respectively, of the set x f x 0 . If we arrange the members of the set x f x 0 in ascending order then we obtain a partition of a, b within which f steps and we see at once that the sum of f over this partition is zero. Since f is zero outside the interval a, b we conclude that f is a step function and that Ý Þ Ý f  0. 4. Given that f is a nonnegative step function and that Ý Þ Ý f  0, prove that the set of numbers x for which f x 0 must be finite. Hint: We begin by choosing an interval a, b and a partition P of a, b such that f x  0 whenever a number x lies outside the interval a, b and such that f steps within the partition P. Now explain briefly why f must have the constant value 0 on each of the open intervals of P. If  j is the constant value of f in the interval x j 1 , x j for each j  1, 2, , n, then, since each number  j is nonnegative we see that for each j, 0 Þa f  b n xi xi 1 i xj xj 1 j i1 from which it follows that each number  j must be zero. 5. Given that f and g are step functions and that c is a real number, prove that 264 0 Ý Ý Þ Ý cf  c Þ Ý f and Ý Þ Ý fg  Ý Ý Þ Ý f  Þ Ý g. These results follow at once when we choose an interval a, b outside of which both f and g are Ý b zero and then replace Þ by Þ . Ý a Exercises on Elementary Sets 1. Given that A and B are elementary sets, prove that m AÞB  m A m B mA B. Solution: We begin by choosing a lower bound a and an upper bound b of the set A Þ B. By looking at the different cases we can see easily that whenever x a, b we have  A ÞB   A   B  A B . Therefore m AÞB   Þ a  A ÞB  Þ a  A   B b b A B Þ...
View Full Document

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.

Ask a homework question - tutors are online