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Maximum-Minimum Theorem, Limits, Continuity and Function Composition

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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

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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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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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 ...

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  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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