# $\{f_n \}$ is equicontinuous and pointwise converge to f is f uniformly continuous?

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.

• What is the domain of the functions? – cmk Jun 4 at 12:30
• Certainly Arzelà–Ascoli theorem implies that $f_n$ is uniformly convergent to $f$, given the domain satisfies some mild conditions. – Tony S.F. Jun 4 at 12:31
• The domain is arbitrary metric space and it is not necessary to be compact. – 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
• So $f_n$ is not uniformly continuous and we know every $f_n$ is uniformly continuous. – amir bahadory Jun 4 at 14:04
• No. None of the $f,f_n$ are uniformly continuous. Shifting does not change the continuity properties. – copper.hat Jun 4 at 14:06