Solved: Bounded functions A real-valued function ƒ is
Chapter 2, Problem 23AAE(choose chapter or problem)
PROBLEM 23AAE
Bounded functions A real-valued function ƒ is bounded from above on a set D if there exists a number N such that ƒ(x) ≥ N for all x in D. We call N, when it exists, an upper bound for ƒ on D and say that ƒ is bounded from above by N. In a similar manner, we say that ƒ is bounded from below on D if there exists a number M such that ƒ(x) ≥ M for all x in D. We call M, when it exists, a lower bound for ƒ on D and say that ƒ is bounded from below by M. We say that ƒ is bounded on D if it is bounded from both above and below.
a. Show that ƒ is bounded on D if and only if there exists a number B such that |ƒ(x)| ≤ B for all x in D.
b. Suppose that ƒ is bounded from above by N. Show that if
c. Suppose that ƒ is bounded from below by M. Show that if
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