Are there any metric space in which every continuous function are bounded but there exists one such function who does not attain its bound.
2026-04-06 12:28:25.1775478505
Metric Space where every real continuous functions are Bounded, but does not attain its Bound.
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Assuming that you are talking about real-valued functions, there is not. A space on which every continuous real-valued function is bounded is called pseudocompact, and every pseudocompact metric space is compact. Finally, every real-valued function on a compact space attains its maximum and minimum values.