Equicontinuity uses one local delta for the whole family; uniform equicontinuity uses one delta everywhere.Suggest a slogan
Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces, let \(\mcf\) be a family of functions \(X\to Y\), and let \(x_0\in X\).
\(\mcf\) is equicontinuous at $x_0$ if and only if for every \(\varepsilon > 0\) there exists \(\delta > 0\) such that \(d_Y(f(x), f(x_0)) < \varepsilon\) for all \(f\in\mcf\) and all \(x\in X\) with \(d_X(x, x_0) < \delta\).
\(\mcf\) is uniformly equicontinuous if and only if for every \(\varepsilon > 0\) there exists \(\delta > 0\) such that \(d_Y(f(x_1), f(x_2)) < \varepsilon\) for all \(f\in\mcf\) and all \(x_1, x_2\in X\) with \(d_X(x_1, x_2) < \delta\).
What is equicontinuity? Uniform equicontinuity?