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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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 Fall '08
 STAFF
 Math, Calculus

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