We can define a limit operator L, that acts on partial functions, as
$$\textrm{L}f(x) = \lim_{y \to x} f(y)$$
So that e.g. continuous functions are fixed points of L. I'm wondering whether this is defined anywhere, and what its properties are.
In particular, we can define an L that extends a function $f : \mathbb{R} \rightharpoonup \mathbb{R}$ to a function $\textrm{L}f : \overline{\mathbb{R}} \rightharpoonup \overline{\mathbb{R}}$, where $\overline{\mathbb{R}}$ is the extended reals $[-\infty, +\infty]$ -- or a function $g : \mathbb{N} \rightharpoonup \mathbb{R}$ to a function $\textrm{L}g : \overline{\mathbb{N}} \rightharpoonup \overline{\mathbb{R}}$, where $\overline{\mathbb{N}}$ is the extended naturals $\mathbb{N} \cup \{\infty\}$ -- such that limits at infinity behave as expected.
In general L doesn't seem to be very well-behaved -- it doesn't preserve composition, isn't idempotent, etc. (though I suppose it is linear). Is there a standard class of space/map that can make L a functor with all the usual properties of infinite limits people expect? If so, what does it do to $\mathbb{Z}$ (homeomorphic to $\mathbb{N}$), $\mathbb{R}^2$, $\mathbb{Q}$? Is it some sort of compactification?