1
$\begingroup$

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

and then use $$\lambda( f_n)$$

1 Answers 1

1

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 $\mathbb{R} \to \mathbb{R}$.

  • 0
    Careful: this is not defined on all functions ${\mathbb R} \to \mathbb R$, just on those that have a limit at $x_0$.2012-09-13
  • 0
    @RobertIsrael: yes, you are of course right. But for smooth functions it doesn't make sense (as one could just evaluate it at $x_0$), and I don't know a symbol for functions which have a limit at $x_0$. I will add a comment in the answer.2012-09-13
  • 2
    It's an unacceptable shorthand, for which the math police will arrest you.2012-09-13
  • 0
    Maybe we can set $D$ as the germ of continuous functions near $x=x_0$.2012-09-13
  • 0
    A function can have a limit at $x_0$ without being continuous anywhere.2012-09-13