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Unformatted text preview: rval p, q within which f steps.
Since f is a step function on the interval p, q , it follows from Exercise 1 that f is a step function on
a, b .
3. Give an example of a step function on the interval 0, 2 that does not step within any regular partition of
0, 2 . Solution: We define
fx 0 if 0
1 if x 2 2 x 2 Now if P is any regular partition of the interval 0, 2 then, since the irrationality of 2 makes it
impossible to find integers n and j such that
2j
2 0 n
we know that 2 can’t be a point of P. In other words, the number 2 must be in one of the open
intervals of P and f fails to be constant in that interval.
4. Explain why a step function must always be bounded.
Suppose that f is a step function on an interval a, b . Choose a partition P of a, b such that f steps
with P. We express P in the form x 0 , x 1 , , x n . Since f is constant in each subinterval x j 1 , x j , the
range of f must be a finite set and therefore f is bounded.
5. Prove that if f and g are step functions on an interval a, b then so are their sum f g and their product fg. Hint: You can find a proof of this assertion in the section on linearity of integration of step functions.
6. Prove that if f and g are step functions then so are their sum f g and their product fg.
Choose an interval a, b such that both of the functions f and g take the value 0 at every number in
R a, b . We deduce from Exercise 2 that both f and g are step functions on the interval a, b and
it follows from Exercise 5 that f g and fg are step functions on a, b and we conclude that these 286 functions are step functions.
7. Prove that a continuous step function on an interval must be constant on that interval.
Suppose that 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
x that P is the partition if x x2 x if x 3 2, 1, 2, 4, 5, 7, 10 of the interval 3 , 2, 10 , that 0 if x 1 1 if 1 x 2 1 if 2 x 4 fx 2 if 4 x5 3 if 5 x7 0 if x 7 3
2
1
0 2 2 4 6 8 10 1 and that Q is the refinement of P given by
Q 2, 1, 2, 3, 4, 5, 6, 7, 10
work out the sums P, f, and Q, f, and verify that they are equal to each other.
We have
P, f, 0 1 1 1 2 1 1 16 2 2 25 16 3 49 25 0 100 49 77 and
Q, f,
1 J , 3 0 1 1 1 2 1 13 2 1 16 9 2 25 16 3 49 25 0 100 49 3 0 1 1 1 2 1 13 2 1 16 9 2 25 16 3 49 25 0 100 49 1 16 9 1 16 2 19 and since
19 3 13 2 we have
Q, f, P, f, 77. 287 2. Given that f is the function whose graph appears in the figure, evaluate Þ Ý
Ý f. 3
2
1
0 2 2 4 6 8 10 1 We sum the function f over...
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 Fall '08
 STAFF
 Math, Calculus

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