Suppose that $f : X \rightarrow Y$ is mapping between topological spaces that is not continuous at $x_0$. Then there is an open set $V$ in $Y$ containing $f(x_0)$ such that for any open set $U$ containing $x_0$, there is some $x_U \in U$ with $f(x_U) \notin V$. By picking one from each $U$ we can build a net $x_U$ converging to $x_0$ such that $f(x_U)$ does not converge to $f(x_0)$. This however requires the axiom of choice because we have to pick $x_U$ from each $U$.

My question is: if we are given that a function between topological spaces is continuous if and only if it is "sequentially" continuous (in the sense of nets, not necessarily infinite sequences indexed by the integers) - does the axiom of choice follow?