If $\{f_n(x)\}$ is an equicontinuous family of functions, does it follow that each function is uniformly continuous? I am a bit confused since in the Arzela-Ascoli theorem, equicontinuity is seems to mean "$\delta$ depends neither on $n$ nor $x$." For example, Wikipedia says:
"The sequence is equicontinuous if, for every $\epsilon > 0$, there exists $\delta>0$ such that
$$|f_n(x)-f_n(y)| < \epsilon$$
whenever $|x − y| < \delta$ for all functions $f_n$ in the sequence."
but I also seem to recall such a term as "uniform equicontinuity." Is the definition not universal?
Best Answer
In the usual statement of Arzela-Ascoli it does not matter, because you are dealing with a compact metric space, where "uniform equicontinuity" is equivalent to "pointwise equicontinuity". (All the functions have a common modulus of continuity at each point, and so an argument exactly parallel to the proof of Heine-Cantor furnishes a uniform modulus of continuity for all of them.) Since this is the main application of the concept, some authors define the word "equicontinuity" to mean what would be more precisely stated as "uniform equicontinuity". Others prefer the more precise term.
The examples I have in mind are Strichartz and Fitzpatrick. Strichartz uses "uniform equicontinuity" everywhere. Fitzpatrick defines "equicontinuity" in the sense of "pointwise equicontinuity" and then makes the equivalence I mentioned an exercise, rather than dwelling on it in his Arzela-Ascoli proof.