Does equicontinuity imply uniform continuity? 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?

Now it does not. The singleton family $F=\{f\}$, with $f(x)=x^2$ is not uniformly continuous, but it is equicontinuous.
However, if $K$ is a compact metric space and $\mathcal F\subset C(K)$ is an equicontinuous family, then $\mathcal F$ is also uniformly equicontinuous.
