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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.

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Welcome to calculus of variations. :) –  Alexander Gruber Oct 27 '12 at 20:12
    
Is there a reason why you take subsequences and not subnets? Is that the point of your question? –  Martin Argerami Oct 28 '12 at 0:50
    
I assume this concept must be pretty basic then haha. I haven't taken topology or calculus. –  user1770201 Oct 28 '12 at 2:08
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