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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 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...
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 Fall '08
 STAFF
 Math, Calculus

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