Let $(X,d)$ be a metric space.
An (intern) horofunction is a function of the form $h\_y:x\mapsto d(x,y)-d(x\_0,y)$, where $x\_0$ is a fixed point.
Now, the map $y\mapsto h\_y$ is one-to-one and continuous (with respect to pointwise convergence topology) so that there is an embedding $X\hookrightarrow \mathrm{Lip}\_{x\_0}(X)$, where $\mathrm{Lip}\_{x\_0}(X)$ is the set of 1-Lipschitz maps $f$ from $X$ to $\mathbb{R}$ such that $f(x\_0)=0$.
Now, (general) horofunctions are elements of the closure of $X$ in $\mathrm{Lip}\_{x\_0}(X)$.


Sometimes, in literature, horofunctions are quotiented by the relation that two functions are the same if they differ by a constant. Actually, it is often convenient to choose a particuliar representative that vanishes at some fixed base point, so the two constructions are quite similar.


In reaction to a comment of Joseph O'Rourke on MO (see [What spaces have well known horofunctions?](https://mathoverflow.net/questions/39885/what-spaces-have-well-known-horofunctions)): does someone know who first introduced horofunctions of that form ?



First introduction of horofunctions
For those who are interested in this, I found a reference in Gromov's book Metric structures for Riemannian and Non-Riemannian spaces (Section 3.11.$\frac{2}{3}\_+$).


Although horofunctions are often attributed to Gromov, they were introduced by Kuratowski in 1935 in the paper "Quelques problèmes concernant les espaces métriques non-séparables" in Fundamenta Mathematicae.


