Let $(f_i)_{i \in I}$ be a directed system of functions $f_j: \mathbb{R} \mapsto \mathbb{R}$ over the directed set $I$. Is there a good way to give a meaning to "direct limit" of this system ?
What are examples where this is possible ?
One can represent an $f_i$ by its graph $\{ (x,f_i(x)) \vert x \in \mathbb{R}\}$, and then apply the classical definition of direct limit, but is there one well-defined way to make the limit into a function $f: \mathbb{R} \mapsto \mathbb{R}$ ?
If you're talking about "directed set" in the same sense as the Wikipedia entry, it makes perfect sense to speak of directed limit.
This is already a well-established concept in topology. E is used to extend the concept of limits of sequences in topological spaces in such a way that analogous theorems for sequences in metric spaces are also valid (with due abstractions) in topological spaces.
See General Topology (Graduate Texts in Mathematics) by John L. Kelley