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Maximum-Minimum Theorem, Limits, Continuity and Function Composition

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]

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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.

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