1873_solutions

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

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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  x1 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  1x1 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  x1 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  x1 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 x1 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  j0 j  xj  xj 1   j1 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...
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