I remember vaguely reading somewhere that limit signs can be pulled out and in if the function was continuous. I forget rigorously what that means, does someone remember what this means?
It was something like:
$$ \lim_{x \to c} f(x) = f(x \to c) = f(x) \to f(c)$$
or was it something like:
$$ f( \lim_{x \to c} x ) = \lim_{f(x) \to f(c)} f(x)$$
I think intuitively what I want to say is that we can bring in f(x) within the limit (which is a special property for some reason) but I can't find the language to express that:
$$ f( \lim x ) = \lim f(x) $$
is the conceptual idea I am trying to express. Any ideas?
Is it just?
$$ f( \lim_{x \to c} x ) = \lim_{x \to c} f(x)$$
Is this what preserving limits means?
If $f$ is a continuous real valued function $f: \mathbb{R} \to \mathbb{R}$ it holds that for every sequence $(x_n)_{n \in \mathbb{N}}$ with $$\lim_{n \to \infty} x_n = a $$ it follows that $$\lim_{n \to \infty} f(x_n) = f(a) $$
which can thus be expressed as
$$ \lim_{n \to \infty} f(x_n) = f(a) = f(\lim_{n \to \infty} x_n)$$