Two different conclusions about continuity implying convergence of sequences - are both true?

Question

I used to think the following was true:

$f$ is continuous, IFF the following holds: Given any sequence in its domain that converges, a corresponding sequence in its codomain converges.

But I recently saw a similar statement:

if $f$ is continuous and $x_n$ converges to $x_0$ in the domain of $f$, then $f(x_n)$ converges to $f(x_0)$.

My question is whether both statements are true. And if so, why is the second statement not an IFF, like the first one is?

Thanks.

EDIT Also, I'm trying to think of a good counter example for the second statement where we have $f(x_n)$ converging to $f(x_0)$ but either $x_n$ does not converge to $x_0$, OR $f$ is not continuous.

Answer 1

It becomes an IFF if you require that for every sequence $x_n$ converging to $x_0$, $f(x_n)$ converges to $f(x_0)$.

Both statements are true.