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?): does someone know who first introduced horofunctions of that form ?
