Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form

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), let $f:X\to Y$ be a function and let $a\in X$.

$f$ is continuous at $a$ if for every real $\epsilon >0$ there is a real $\delta >0$ such that

$d_X(x,a)<\delta ⟹d_Y(f(x),f(a))<\epsilon \text{for all }x\in X\mathrm{.}$

$f$ is continuous (globally, or on $X$) if it is continuous at every point of $X$.

The same condition in balls. Since $d_X(x,a)<\delta$ says $x\in B_X(a,\delta )$ and $d_Y(f(x),f(a))<\epsilon$ says $f(x)\in B_Y(f(a),\epsilon )$ (Open ball, closed ball and sphere in a metric space), continuity at $a$ reads: for every $\epsilon >0$ there is $\delta >0$ with

$f[B_X(a,\delta )]\subseteq B_Y(f(a),\epsilon )\mathrm{.}$

Both forms are used below and are the same statement written twice.

Both metrics matter, and both are named. Continuity is a property of the triple $(d_X,d_Y,f)$, not of $f$ alone. When several metrics on the same underlying sets are in play, as in Topologically, uniformly and Lipschitz equivalent metrics on a set, the metrics are always written out.

Quantifier order. The $\delta$ is allowed to depend on $\epsilon$ and on the point $a$. Requiring one $\delta$ to work at every point simultaneously is a strictly stronger condition, uniform continuity; it is defined on a later page of this library, and at this point in the reading order it is written out in full where needed (Topologically, uniformly and Lipschitz equivalent metrics on a set).

• Nothing is claimed here beyond the definition. Without any choice principle, continuity is equivalent to preimages of open sets being open, to preimages of closed sets being closed, and to $f[\bar{A}]\subseteq \overline{f[A]}$; each of these conditions implies sequential continuity. Assuming Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), sequential continuity is equivalent to them as well. These statements are proved in Metric continuity characterisations, with countable choice for the sequential converse.
• Continuity at a point is a local condition: it depends only on the values of $f$ on any one ball around $a$, since the condition may always be tested with a smaller $\delta$.
• Every isometric embedding is continuous, with $\delta =\epsilon$ (Isometry, isometric embedding, and the subspace metric on a subset, An isometric embedding is injective and carries the metric topology of the source onto the subspace topology of its image), and so is every map that does not increase distances, such as $u\mapsto d(u,A)$ for a fixed nonempty subset $A\subseteq X$ ($\mathrm{\mid }d(x,A)-d(y,A)\mathrm{\mid }\le d(x,y)$, so the distance to a fixed nonempty set is $1$-Lipschitz).