Let $\{f_n \}$ is equicontinuous and pointwise converge to $f$ now is $f$ uniformly continuous ? I can prove f is continuous but is this uniformly continuous? and we know every $f_n$ is uniformly continuous. Is $ f_n$ uniformly convergent to $f$ ? prove or disprove.

  • $\begingroup$ What is the domain of the functions? $\endgroup$ – cmk Jun 4 at 12:30
  • $\begingroup$ Certainly Arzelà–Ascoli theorem implies that $f_n$ is uniformly convergent to $f$, given the domain satisfies some mild conditions. $\endgroup$ – Tony S.F. Jun 4 at 12:31
  • $\begingroup$ The domain is arbitrary metric space and it is not necessary to be compact. $\endgroup$ – amir bahadory Jun 4 at 12:32

Take any $f$ that is not uniformly continuous.

Let $f_n = f+{1 \over n}$.

Then the $f_n$ are equicontinuous and converge pointwise to $f$ which is not uniformly continuous

  • $\begingroup$ So $f_n$ is not uniformly continuous and we know every $f_n$ is uniformly continuous. $\endgroup$ – amir bahadory Jun 4 at 14:04
  • $\begingroup$ No. None of the $f,f_n$ are uniformly continuous. Shifting does not change the continuity properties. $\endgroup$ – copper.hat Jun 4 at 14:06
  • $\begingroup$ Uniform continuity and equicontinuity are different things. $\endgroup$ – copper.hat Jun 4 at 14:06
  • $\begingroup$ Equicontinuity implies uniform continuity of every member of the family. You say this is false? $\endgroup$ – amir bahadory Jun 4 at 14:14
  • $\begingroup$ Equicontinuity is a local property that applies to all members of the family, uniform continuity is a global property of a single function. $\endgroup$ – copper.hat Jun 4 at 14:17

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