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Real Analysis : Bounded Continuity / Differentiability

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Let f: [0, ∞) → R be a bounded function. For all X greater than or equal to 0, let G(x)=sup{f(t): 0 is less than or equal to t is less than or equal to x}

a) Show that if f is continuous, g is also continuous. Is the converse also true? Justify.

b) If f is differentiable and continuous, is g also differentiable and continuous? Justify.

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Bounded Continuity and Differentiability are investigated. The solution is detailed and well presented.

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is a bounded function. For each , let .
1. Show that if is continuous, then is continuous. Is the converse also true?
If is continuous, then for each , is continuous on the closed interval . So can get the maximum value ...

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