Let $\{f_j\}$ be a family of self-homeomorphisms of $\overline{\mathbb{R}}^n$ and $x,y\in\overline{\mathbb{R}}^n$, where $\overline{\mathbb{R}}^n$ is the one-point compactification of $\mathbb{R}^n$. What does it mean for $\{f_j\}$ to converge c-uniformly to y in $\overline{\mathbb{R}}^n\setminus \{x\}$? That is, what does "$\lim_{j\rightarrow\infty} f_j =y$, c-uniformly in $\overline{\mathbb{R}}^n\setminus {x}$" mean? I'm only confused about what function the point $y$ represents. I would have assumed it meant the function that sends everything to the point $y$, but this is not a bijection on $\overline{\mathbb{R}}^n$.
The source for this question is the paper "Discrete Quasiconformal Groups I" by Gehring and Martin, Proc. London Math. Soc. (3) 55, 1987. Definition 3.3 and Theorem 3.7. My interest here is in understanding convergence groups, in particular their use in defining relatively hyperbolic groups.