1873_solutions

Now fill in the details why f must be strictly

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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 x5 3 if 5 x7 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 i1 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 fg  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...
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