Do continuous functions "preserve limits" between spaces where uniqueness of limits isn't guaranteed? As far as I know (I think I am not mistaken), the following is true: "Given two Hausdorff spaces $X$ and $Y$ and $f$ a mapping between them, the following two statements are equivalent:

*

*$f$ is continuous

*Given a sequence $(x_n)$ converging to $x$, then the sequence $(f(x_n))$ converges to $f(x)$"

My question is: can this be generalized in the case of "convergence without uniqueness"? The proposed generalization being:
"Given two topological spaces $X$ and $Y$, a mapping $f$ between them is continuous if and only if, for any sequence $(x_n)$ converging to a set of points $A$, the corresponding sequence $(f(x_n))$ is convergent, and converges to the set $f(A)$"
I specially care about the "only if" part, because I need it for further results that I'm trying to achieve. Of course, I already come with a proof for this part that I think is correct, but I ask here this question to put this proof to the test of collective scrutiny. The proof in question:
"Let $X$ and $Y$ be topological spaces, let $f$ be a continuous mapping between them. Let $(x_n)$ be a sequence converging to a set $A$, let $x$ be a point in A. Then for each open $G$ containing x, there exists a certain $n_0$ such that for any $n > n_0$, $x_n$ is in $G$.
Let $H$ be an open in $Y$ such that $f(x)$ is in $H$. Then $f^{-1}(H)$ is an open in $X$ containing $x$. Then there exists $n_0$ with the above property for $f^{-1}(H)$. But
$$x_n\in f^{-1}(H) \Leftrightarrow f(x_n) \in H$$
Then $f(x)$ is a limit point of the sequence $(f(x_n))$."
Is this proof correct? Also, is the other implication true? Thank you for your attention.

The converse statement is false. Let $\tau_{\rm{usual}}$ be the usual topology on $\mathbb R$ and $\tau_{cc}$ be the co-countable topology on $\mathbb R$, that is :
$$\tau_{cc} = \{U\subset \mathbb R~|~U= \emptyset \text{ or } X\backslash U\text{ is at most countable} \}$$
Then, $\tau_{cc}$ is not Hausdorff and a sequence in $(\mathbb R , \tau)$ is convergent if, and only if, it is eventually constant.
Therefore, the identity $\text{id}:(\mathbb R,\tau_{cc}) \to (\mathbb R, \tau_{\rm{usual}})$ sends convergent sequence to convergent sequences (and limit points to limit points) but is not continuous.
