It is known that the complement of an analytic set is connected. In general, the complement of a proper complex analytic set in a connected complex manifold is an arcwise connected dense open set. My question is this:

If $f: X \rightarrow Y$ is an analytic mapping, where $X$ and $Y$ are connected complex manifolds, and if $A \subseteq X$ is an analytic set, is it necessarily true that the complement of $f(A)$ in $Y$ is connected?

If necessary, assume $X$ and $Y$ have the same dimensions, or even $X=Y=\mathbb{C}^n$ (the answer is yes, when $n=1$).

It is known that the complement of an analytic set is connected. In general, the complement of a proper complex analytic set in a connected complex manifold is an arcwise connected dense open set. My question is this:

If $f: X \rightarrow Y$ is an onto analytic mapping, where $X$ and $Y$ are connected complex manifolds, and if $A \subseteq X$ is an analytic set, is it necessarily true that the complement of $f(A)$ in $Y$ is connected?