Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces, let \(\mcf\) be a family of functions \(X\to Y\), and let \(x_0\in X\).
The family \(\mcf\) is equicontinuous at \(x_0\) 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 \(d_X(x, x_0) < \delta\).
The family \(\mcf\) is uniformly equicontinuous 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 \(d_X(x_1, x_2) < \delta\).
In equicontinuity at \(x_0\) the number \(\delta\) depends on \(\varepsilon\) and \(x_0\) but not on \(f\in\mcf\); in uniform equicontinuity one \(\delta\) depending only on \(\varepsilon\) serves every \(f\in\mcf\) and every pair of points of \(X\).