Reversed continuity We know that if a function $f$ satisfies
$$\lim_{n\rightarrow\infty}f(x_n) = f(x_0),\qquad (1)$$
where $\lim_{n\rightarrow\infty}x_n = x_0$, than $f$ is continuous at the point $x_0$. But can we reverse this statement? What I mean by that is, if we already know that $f$ is continuous at $x_0$, can we say that $(1)$ must be true?
 A: Your statement is wrong ! Take $f=\boldsymbol 1_{\left\{\frac{1}{n}\mid n\in\mathbb N\right\}}$ and $x_n=\frac{\pi}{n}$. Then $f(x_n)\to f(0)$, but $f$ is not continuous at $0$.
The statement should be :

$f$ is continuous at $x_0$ if and only if for all $(x_n)$ s.t. $x_n\to x_0$, we have $$\lim_{n\to \infty }f(x_n)=f(x_0).$$

A: Yes, this is a definition, which should be interpreted as an "if and only if" statement rather than an "if" statement, even though definition are usually written using only "if". So $f$ is continuous at the point $x_0$ if and only if whenever $\lim_{n \to \infty} x_n = x_0$, we have $\lim_{n \to \infty} f(x_n)=f(x_0)$.
For another example, you'll often see the definition "$n$ is even if $n=2k$ for some integer $k$". But really $n$ is even if and only if $n=2k$ for some integer $k$. Whenever you are dealing with definitions, you should think of the condition as an "if and only if".
A: There is well known definition In terms of sequences by Eduard Heine, by which function is continuous in $x_0$ when for $\forall x_n \to x_0, \exists N, n>N, x_n \ne x_0$ (your initial statement leaks conditions on $x_n$) we have $\lim_{n\rightarrow\infty}f(x_n) = f(x_0)$ and it is equivalent to classical Cauchy $\epsilon - \delta$  definition.
So if function is continuous by Cauchy, then your (1) holds.
