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Is it OK to set something like : $\lambda := \lim _{x\rightarrow x_0}$

and then use $\lambda( f_n)$

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I think, a more conventional way would be to define a mapping $\lambda: (\mathbb{R} \to \mathbb{R}) \to \mathbb{R}$ with $\lambda(f) = \lim_{x\to x_0} f(x).$

Edit: as Robert Israel pointed out, one should of course make clear that the mapping is only defined for functions where the limit exists. So $\lambda: D \to \mathbb{R}$ where $D$ is the subset of $\mathbb{R} \to \mathbb{R}$.

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    A function can have a limit at $x_0$ without being continuous anywhere.2012-09-13