Unformatted text preview: n N which we can write as
n xn n xN
N . Since
we deduce that no partial limit of the sequence n x n can be more than and, therefore, that the number
u cannot be a partial limit of the sequence n x n .
n Ýn 11 The Riemann Integral
Some Exercises on Step Functions
1. True or false? If f is a step function on an interval a, b and c, d is a subinterval of a, b then f is a step
function on c, d . Solution: The statement is true.
Choose a partition P of a, b within which the function f steps and then refine P by adding to it the two
numbers c and d. Then drop all of the points of this partition that lie outside of the interval c, d and we
obtain a partition Q of c, d within which f steps.
2. True or false? If f is a step function then given any interval a, b , the function f is a step function on a, b . Hint: This statement is true. Write a short proof. Choose an interval c, d such that f is a step function on c, d and such that f x 0 whenever
x R c, d . Choose numbers p and q such that p is less than both of the numbers a and c and q
is greater than both of the numbers b and d p a c b d q Choose a partition P of the interval c, d such that f steps within P. If we add the two numbers p and
q to the partition P then we obtain a partition of the larger interval 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 262 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
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
functions are step functions.
7. Prove that a continuous step function on an interval must be constant on that interval.
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