Is a uniformly continuous function always continuous? Does continuity imply uniform continuity?
I'm struggling a bit in deciphering the difference between these. Can you strategically prove the assertions?
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5Everyone struggles with this; this means that the question has been asked and answered here many times before. Try How exactly can't δ depend on x in the definition of uniform continuity? or the questions in the "Linked" column of that one, or try a search for "continuous uniformly-continuous difference". – MJD Feb 18 '15 at 03:47
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1The short answers are: uniformly continuous functions are always continuous, but not vice versa in general. – Greg Martin Feb 18 '15 at 04:35
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Thanks MJD! Read through that link and understood the concept much better :) – BetaGammaTao Feb 18 '15 at 18:39
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Any function f that is uniformly continuous is also continuous. However, not all continuous functions are uniformly continuous. However, if we have a compact metric space and our function f is continuous, then f is also uniformly continuous.
Mike El Jackson
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