2.50 Prove: A function f is bounded if and only if || f |lsup < o. (Hint: See Definition 2.4.1.) 2.4 EXTREME VALUE THEOREM AND ITS CONSEQUENCES Definition 2.4.1 We say a function f is bounded on a domain D, provided that there exist m and M in R such that m < f(x) < M for all x E D. Observe that f(x) = x is continuous on R but f is not bounded on R. And g(x) = ! is continuous on (0,1) but is not bounded, since it lacks an upper bound, though there is a lower bound, 0. We have, however, the following important theorem pertaining to continuous functions on closed, finite intervals.

2.50 Prove: A function f is bounded if and only if || f |lsup < o. (Hint: See Definition 2.4.1.) 2.4 EXTREME VALUE THEOREM AND ITS CONSEQUENCES Definition 2.4.1 We say a function f is bounded on a domain D, provided that there exist m and M in R such that m < f(x) < M for all x E D. Observe that f(x) = x is continuous on R but f is not bounded on R. And g(x) = ! is continuous on (0,1) but is not bounded, since it lacks an upper bound, though there is a lower bound, 0. We have, however, the following important theorem pertaining to continuous functions on closed, finite intervals.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 63RE

See similar textbooks