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.