What is the target space? If . Trivially no. Example: .

For (say). Trivially yes.
The reason is, let be open and . Let’s say . Then it is easy to see that is nonconstant OR is non-constant (by Taylor’s expansion, say). Say is non-constant, then it maps a small (1-dim!) neighborhood of to an open set around , by 1-D open mapping theorem. So there is an open set around which is contained in .[My argument is wrong. ]

It turns out that I can correct my argument. Let my clarify it here.
For convenience, let’s assume be a non-constant holomorphic (entire) function. Let be open and . Wlog, we assume and
. We claim that there is an open set which contains . To see this, since is non-constant, there is with . Define to be an entire non-constant holomorphic function. Then by open mapping theorem, maps arbitrary small neighborhood of onto an open set around . So by choosing this neighborhood to be small enough, we see that there is an open set containing .

Zorn:

What is the target space? If . Trivially no. Example: .

~~For (say). Trivially yes.~~[My argument is wrong. ]The reason is, let be open and . Let’s say . Then it is easy to see that is nonconstant OR is non-constant (by Taylor’s expansion, say). Say is non-constant, then it maps a small (1-dim!) neighborhood of to an open set around , by 1-D open mapping theorem. So there is an open set around which is contained in .

It turns out that I can correct my argument. Let my clarify it here.

For convenience, let’s assume be a non-constant holomorphic (entire) function. Let be open and . Wlog, we assume and

. We claim that there is an open set which contains . To see this, since is non-constant, there is with . Define to be an entire non-constant holomorphic function. Then by open mapping theorem, maps arbitrary small neighborhood of onto an open set around . So by choosing this neighborhood to be small enough, we see that there is an open set containing .