TITLE: Spaces with the property: Uniformly continuous equals continuous
QUESTION [6 upvotes]: I found a nice book about functional analysis with a nice theorem in it: Continuity at 0 is equal to Lipschitz continuous for linear maps in normed spaces.
This fact inspires me to ask: Are there spaces with the property: uniformly continuous is the same as continuous (everywhere)?
My problem is to find (metric) spaces with this property, so that I'm able to find uniform structure on topological spaces.
Is there a name for this theory?

REPLY [5 votes]: Such metric spaces are called UC-spaces (Google Scholar, Google Books) or Atsuji spaces. (Google Scholar, Google Books).
Quote from G. Beer: Topologies on Closed and Closed Convex Sets, p.54:

2.3.1 Theorem. Let $\langle X,d \rangle$ be a metric space. The following are equivalent:
  (1) Each continuous function on $\langle X,d\rangle$ with values in an arbitrary metric space $\langle Y.d'\rangle$ is uniformly continuous;
  (2) Each real-valued continuous function on $\langle X,d\rangle$ is uniformly continuous;
  (3) Whenever $A$ and $B$ are disjoint elements of $\operatorname{CL}(X)$, then there exits $\varepsilon>0$ such that $S_\varepsilon[A]\cap S_\varepsilon[B]=\emptyset$;
  (4) The set $X'$ of accumulation points of $X$ is compact, and $\forall \varepsilon>0$ such that $S_\varepsilon[X']^c$ is uniformly discrete: $\exists\delta>0$ such that if $x\ne w$ and $\{x,w\}\subset S_\varepsilon[X']^c$ then $\delta>0$;
  (5) Whenever $\langle x_n\rangle$ is a sequence in $X$ with $\lim_{n\to\infty} d(x_n,\{x_n\}^c)=0$, then $\langle x_n\rangle$ has a cluster point:
  (6) Each open cover of X has a Lebesgue number: there exist a number $\lambda > 0$ such that each subset of $X$ of diameter at most $\lambda$ lies entirely in one member of the cover.

Here $\operatorname{CL}(X)$ denotes the set of all non-empty closed subsets of $X$ and $S_\varepsilon[A]=\{x\in X \colon d(x,A)<\varepsilon\}$.

Some authors also use name Lebesgue spaces, see the comments here.
Another related question is: Inverse of Heine–Cantor theorem