Sequences, subsequences, and continuity of functions

Question

It's been a few years since I studied point-set topology, and I'm a bit rusty on the basics. Would appreciate help with the following question.

Suppose $f:X\rightarrow Y$ is a map between two topological spaces, and I know that for any sequence $x_n\rightarrow x$ in $X$, there is a subsequence $x_{n_k}$ such that $f(x_{n_k})\rightarrow f(x)$. Does it follow that $x_n\rightarrow x\Rightarrow f(x_n)\rightarrow f(x)$? In the case when $X$ is first countable, the first condition is enough to imply that $f$ is continuous (I believe), and so the second condition must hold. I suspect this isn't true for general $X$, but I can't come up with a counterexample.

Thanks.

Answer 1

If $x_n\to x$ and $f(x_n)$ doesn't converge to $f(x)$, take $V$ a neighborhood of $f(x)$ such that for infinitely many $k$, $f(x_k)\notin V$. We can write this infinitely many as a subsequence $x_{n_k}$. We have $x_{n_k}\to x$, but we can't extract a subsequence of $\{f(x_{n_k})\}$ which is convergent to $f(x)$.