Suppose $\{f_i\}$ is an uncountable family of functions from $X$ to $\mathbb{R}$.
If $\exists f: X \rightarrow \mathbb{R}$ where $\forall \{f_{i_n}\} \subseteq \{f_i\}$ s.t. $f_{i_n} \ne f$ and $f_{i_n} \rightarrow f$ pointwise, is there a special name for $f$?
It seems that $f$ is acting as the functional analogue of a limit point in a sequence of real numbers.