The question is problem 2-15 in "Introduction to Topological Manifolds" by John. M Lee:
Let $X$ and $Y$ be topological spaces.
a) Suppose $f: X \to Y$ is continuous and $p_n \to p$ in $X$. Show that $f(p_n) \to f(p)$ in $Y$.
b) Prove that if $X$ is first countable, the converse is true: if $f: X \to Y$ is a map such that $p_n \to p$ in $X$ implies $f(p_n) \to f(p)$ in $Y$ , then $f$ is continuous.
For a), if we take a neighborhood $V$ of $f(p)$, we know that $f^{-1}(V)$ is open in $X$, since $f$ is continuous. But do we know that $p\in f^{-1}(V)$, i.e. that $f^{-1}(V)$ is a neighborhood of $p$?
For b) I am completely lost and could use some help.
Thanks!
