There is a theorem that states that every continuous function from a compact metric space to R is bounded however, the converse is not true, but I am not sure how to construct such a function. My idea is to start with a countable subcover that has no finite subcover. Since we can enumerate each subcover U1, U2, U3..., I was thinking of mapping each element in the subcover equal to its index, but I'm not sure if this function is continuous.
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The converse is true. – José Carlos Santos Apr 12 '21 at 10:29
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sorry, i meant i was just trying to construct a continuous function from a noncompact metric space to R that is unbounded – Bill Apr 12 '21 at 10:31