Equicontinuity at a point, uniform equicontinuity, and pointwise boundedness of a family of maps between metric spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and let $\mathcal{F} \subseteq Y^{X}$ be a set of functions $X \to Y$ (The topology of pointwise convergence on $Y^{X}$, which is the product topology, and its restriction to $C(X,Y)$). Let $a \in X$.

• $\mathcal{F}$ is equicontinuous at $a$ if for every real $\varepsilon > 0$ there is a real $\delta > 0$ such that $d_Y\big(f(x), f(a)\big) < \varepsilon \qquad \text{for every } f \in \mathcal{F} \text{ and every } x \in X \text{ with } d_X(x,a) < \delta .$
• $\mathcal{F}$ is equicontinuous if it is equicontinuous at every point of $X$.
• $\mathcal{F}$ is uniformly equicontinuous if for every real $\varepsilon > 0$ there is a real $\delta > 0$ such that $d_Y\big(f(x), f(x')\big) < \varepsilon \qquad \text{for every } f \in \mathcal{F} \text{ and all } x, x' \in X \text{ with } d_X(x,x') < \delta .$
• $\mathcal{F}$ is pointwise bounded if for every $x \in X$ the set $\mathcal{F}(x) := \{\, f(x) : f \in \mathcal{F} \,\}$ is a bounded subset of $Y$ (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

Everything is in the quantifier order, and the order is the only difference from ordinary continuity. Continuity of each single $f \in \mathcal{F}$ at $a$ allows $\delta$ to depend on $\varepsilon$, on $a$ and on $f$ (Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form); equicontinuity at $a$ demands one $\delta$ serving every member of the family at once. Uniform continuity of each single $f$ allows $\delta$ to depend on $\varepsilon$ and on $f$ (Uniform continuity of a map of metric spaces: one $\delta$ serving every point); uniform equicontinuity demands one $\delta$ serving every member and every pair of points at once. Written with the quantifiers in order, the four conditions are

$\forall \varepsilon\, \forall f\, \forall a\, \exists \delta, \qquad \forall \varepsilon\, \forall a\, \exists \delta\, \forall f, \qquad \forall \varepsilon\, \forall f\, \exists \delta\, \forall a, \qquad \forall \varepsilon\, \exists \delta\, \forall f\, \forall a$

for pointwise continuity of each member, equicontinuity, uniform continuity of each member, and uniform equicontinuity respectively.

Immediate consequences, recorded because they are used.

1. Every member of an equicontinuous family is continuous, and every member of a uniformly equicontinuous family is uniformly continuous: the $\delta$ that serves the whole family serves each member (Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form, Uniform continuity of a map of metric spaces: one $\delta$ serving every point).
2. Uniform equicontinuity implies equicontinuity, by taking $x' = a$.
3. Both conditions are about the metrics $d_X$ and $d_Y$, not about the topologies they induce. Replacing a metric by a topologically equivalent one can destroy either, exactly as it can destroy uniform continuity.
4. A one-element family $\{f\}$ is equicontinuous exactly when $f$ is continuous, and uniformly equicontinuous exactly when $f$ is uniformly continuous; so the notions do generalise the single-function ones and do not merely resemble them.

Pointwise boundedness is a hypothesis about the values, not about the functions. It says that at each individual point the family's values stay in one ball of $Y$; the radius may depend on the point, and no single ball need contain all the values at all the points. The stronger condition, that $\bigcup_{x} \mathcal{F}(x)$ is bounded, is uniform boundedness and is not defined here, nothing on this page using it.

• Why this definition sits on this page. Equicontinuity is the hypothesis of the Ascoli-Arzelà theorem, which characterises the compact subsets of $C(X,Y)$ in the topology of compact convergence (The topology of compact convergence on $C(X,Y)$ for metric $X$ and $Y$: uniform convergence on each compact subset of $X$). That theorem is not proved here and is not stated here; the definition is placed on this page so that the page proving it has the vocabulary available earlier in the reading order. Nothing below this item uses equicontinuity except the companion page's examples.

• Neither condition is implied by the other two hypotheses of Ascoli. Pointwise boundedness does not imply equicontinuity, and the companion page gives a family of continuous functions on $[0,1]$ with all values in $[0,1]$ that fails to be equicontinuous at $0$. Conversely an equicontinuous family need not be pointwise bounded: the constant functions with values $0, 1, 2, \dots$ are uniformly equicontinuous and unbounded at every point.

• A convenient sufficient condition. A family of maps that are all Lipschitz with one common constant $L$ is uniformly equicontinuous, $\delta := \varepsilon/(L+1)$ serving. The companion page uses this for the $1$-Lipschitz maps into $\mathbb{R}$, among them all the distance functions $x \mapsto d_X(x,A)$.