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Suppose f: K->(-
,), K is compact, and f has a finite limit at each point of K, but may not be continuous on K. Show that f is bounded in two ways: (i) by using the definition of compactness in terms of open covers, and (ii) by using the sequential characterization of compactness. Is the same conclusion valid if we drop the assumption that the limit of f is finite?Let a,b be real numbers with a < b.
(1) Give an example of a continuous f on [a,b) which is not bounded.
(2) Give an example of a continuous f on [a,infinity) which is not bounded.
(3) Give an example of a bounded f on [a,b] for which sup_[a,b] f is not achieved.
(4) Give an example of a bounded, continuous f on [a,infinity) for which sup_[a,infinity) f is not achieved.
(5) Give an example of a bounded, continuous f on [a,b) for which sup_[a,b) f is not achieved.
I just get so confused, I feel like i need to see the examples to understand them.
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