Explore BrainMass

Explore BrainMass

    Maximum-Minimum Theorem, Limits, Continuity and Function Composition

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    1) Let f, g be defined on R and let c in R. Suppose that lim f = b and that g is continuous at b. Show that lim g 0 f = g(b)

    Note:
    R: real numbers
    g 0 f means composition of f and g

    2) Let A = [0, 1) U (1,2]. Let B = [0, 1] U [2, 3]. Does the conclusion of the maximum-minimum theorem always hold for a function f: A -> R, g: B ->R that is continuous on A, on B respectively? Prove or give a counterexample.

    Note:
    U: means union
    Max-min theorem: f: [a, b] ->R continuous on [a, b]. Then f has an absolute max and an absolute min on [a, b]

    © BrainMass Inc. brainmass.com March 4, 2021, 6:20 pm ad1c9bdddf
    https://brainmass.com/math/linear-programming/maximum-minimum-theorem-limits-continuity-40412

    Solution Preview

    Please see the attached file for the complete solution.
    Thanks for using BrainMass.

    1) Let f, g be defined on R and let c in R. Suppose that lim f = b and that g is continuous at b. show that lim g 0 f = g(b)

    Note:
    R: real numbers
    g 0 f means composition of f and g

    Proof. Assume that , we want to show that

    It suffices to show that, for , there exists ...

    Solution Summary

    Maximum-Minimum Theorem, Limits, Continuity and Function Composition are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

    $2.49

    ADVERTISEMENT