What does it mean for a function to "preserve the limits of sequences"?

Question

I've translated the English Wikipedia page "Limit of a sequence". What does the following statement mean?

In fact, any real-valued function f is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity).

I cannot understand the following part: "... it preserves the limits of sequences ...". Can you write same sentence in a long and clear way? Thank you in advance.

Answer 1

Here is what they mean. $f$ is continuous at $a \in \mathbb{R}$ if and only if for any sequence $x_1, x_2, x_3, \ldots,$, if $x_n$ converges to $a$, then $f(x_n)$ converges to $f(a)$.

When you say $x$ preserves $y$, that means, "$x$ does not change $y$", or "$x$ keeps $y$ the same". In mathematics if you say that a function $f$ preserves some property $P$, that usually means that if you apply $f$, property $P$ is still true. So if a function $f$ preserves the limits of sequences, that means that every sequence limit ($x_n \to a$) is still true when you apply $f$: $f(x_n) \to f(a)$.

Answer 2

I think they simply mean that $f$ is continuous (at $x_0$) iff

$$ \lim\limits_{x\to x_0} f(x) = f(\lim\limits_{x\to x_0} x) = f(x_0). $$

Using sequences then for any sequence $(x_n)_{n\in\mathbb{N}}$ where $x_n\to x_0$ as $n\to\infty$, we have that

$$ \lim\limits_{n\to\infty} f(x_n) = f(\lim\limits_{n\to\infty} x_n) = f(x_0). $$

By the word 'preserves', you may try to replace it with 'conserves', 'keeps', 'retains', or 'upholds'.

Answer 3

In plain English:

A real-valued function $f$ is continuous if, for any sequence that converges, the image by $f$ of the sequence converges, and the limit of the image is the image of the limit.