1873_solutions

F 0 f 1 a b the case b f a in this case the number

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: the partition P of the interval 2, 10 . Ý Þ0 f  0 1 2 1 2 2, 1, 2, 4, 5, 7, 10 1 3. In each of the following cases evaluate Þ Ý Ý 2 2 5 14 4 3 7 5  0 10 fd. a. We define 2 if fx  x1 1 0 if x R 1, 1 and 0 if x  1 x  Ý 3 if x Þ Ý fd  Þ 1 fd  2 3 1 1 . 0 2 3 3 6 b. We define 2 if fx  1x1 0 if x R 1, 1 and 0 if x  1 x  Ý 3 if x Þ Ý fd  Þ 1 fd  0 3 1 1 . 0 2 3 3 0 c. We define 2 if fx  x1 1 0 if x R 1, 1 and x  Ý 0 if x 3 if x  1 Þ Ý fd  Þ 1 fd  2 3 1 288 1 . 0 2 3 3 6 7  7. d. We define 2 if fx  x1 1 0 if x R 1, 1 and 0 if x  1 x  2 if x  1 . 3 if x Ý Þ Ý fd  Þ 1 fd  2 3 1 1 0 2 3 3 6 e. We define 2 fx  if 0 x1 1 if 2 x if x R 0 6 0, 1 Þ 2, 6 and 5 if x x  0 Ý if 0  x  3 x 0 0 if x 1 2 13 0 2 0 3 3 6 Þ Ý fd  Þ 0 fd 6 20 5  0 10 0 16 3  4. 4. Prove that if  is an increasing function and f is a step function then the function |f | is a step function and Ý Ý Þ Ý fd Þ Ý |f |d. The fact that |f | is a step function whenever f is a step function follows at once from the fact that |f | is constant on any interval on which the function f is constant. Since |f | f |f |, it follows from nonnegativity that Ý Ý Ý Þ Ý |f |d Þ Ý fd Þ Ý |f |d and we conclude that Ý Þ Ý fd Ý Þ Ý |f |d. 5. Given that f is a function defined on R and that the set of numbers x for which f x 0 is finite, explain why f must be a step function and why if  is a continuous increasing function we must have Ý Þ Ý fd  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. Since the jump of  at each of the numbers in the set x f x 0 is zero, 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. 289 6. Given that f is a nonnegative step function and that  is a strictly increasing function and that Ý Þ Ý fd  0, prove that the set of numbers x for which f x 0 must be finite. We begin by choosing an interval a, b and a partition P  x 0 , x 1 , , x n 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. If the constant value of f in each interval x j 1 , x j is  j then n n f x j J , x j  j0 j  xj  xj 1   j1 Ý Þ Ý fd  0 and, since every term in this summation is nonnegative we know that every term must be zero. For each j, the fact that  is strictly increasing and the fact that j  xj  xj 1  0 guarantees that  j  0. 7. Given that f and g are step functions, that  is an increasing function and that c is a real number, prove that Ý Ý Þ Ý cfd  c Þ Ý fd and Ý Ý Ý Þ Ý f  g d  Þ Ý fd  Þ Ý gd. 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...
View Full Document

Ask a homework question - tutors are online