In the literature I find several definitions for (uniform) equicontinuity of a family $M\in C[0,1]$. $M$ is called (uniformly) equicontinuous if ...
(1) $\forall\varepsilon>0\;\exists\delta>0$ s.t. $|x(t)-x(s)|<\varepsilon$ for all $x\in M$ and $t,s \in[0,1]$ with $|t-s|<\delta$.
(2) $\forall\varepsilon>0\;\exists\delta>0$ s.t. $\sup_{x\in M}|x(t)-x(s)|<\varepsilon$ for all $t,s \in[0,1]$ with $|t-s|<\delta$.
(3) $\forall\varepsilon>0\;\exists\delta>0$ s.t. $\sup_{x\in M}\sup_{t,s\in [0,1], |t-s|<\delta}|x(t)-x(s)|<\varepsilon$.
(4) $\lim_{\delta\to 0}\sup_{x\in M}\sup_{t,s\in [0,1], |t-s|<\delta}|x(t)-x(s)|=0$.
Are these definitions all equivalent?