Continuity on Metric Spaces

Nothing in the definition of continuity used the arithmetic of $\mathbb{R}$; it used only the distances $|x-c|$ between inputs and $|f(x)-f(c)|$ between outputs. Replace those absolute values with a metric on the domain and a metric on the codomain, and the theory of real limits and continuity acquires a general form. The abstraction covers functions of several variables, maps between function spaces, and operators on sequences under one definition.

Continuity between metric spaces

Read it as before: every output ball of radius $ϵ$ around $f(c)$ is reached by some input ball of radius $\delta$ around $c$. On $\mathbb{R}$ with the standard metric this reduces to the old definition. The sequential characterization carries over unchanged.

Compactness and the extreme value theorem

The proof that a continuous function attains its extremes on $[a,b]$ used only that $[a,b]$ is compact. In a metric space the same argument, via sequential compactness, gives the general statement.

The statement about closed bounded intervals is the special case $X=[a,b]$.

Continuity likewise preserves connectedness: the image of a connected space under a continuous map is connected. This is the general intermediate value theorem: if $X$ is connected and $f:X\to \mathbb{R}$ is continuous, $f$ takes every value between any two of its values.

The topological characterization

The formulation that generalizes most cleanly drops $ϵ$ and $\delta$ and speaks only of open sets.

Uniform continuity and homeomorphisms

The definition of uniform continuity transfers the same way.

As on the line, Lipschitz maps (those with $d_Y(f(p),f(q))\le K{d}_X(p,q)$) are uniformly continuous, with $\delta =ϵ/K$.

The open-set formulation also defines the maps that identify two metric spaces topologically.

A continuous bijection need not have a continuous inverse in general, but compactness forces it: a continuous bijection from a compact space to a metric space is automatically a homeomorphism, since it carries closed (hence compact) sets to compact, hence closed, sets, so preimages under $f^{-1}$ of closed sets are closed. Homeomorphisms preserve every property defined purely from the open sets, compactness and connectedness among them.

Two settings on the real line fall outside this metric picture: limits as the variable runs off to $\pm \infty$, and the control that monotonicity imposes on where a function may be discontinuous. Both use the order of $\mathbb{R}$, which a general metric space lacks.