What I had in mind was something like the following:

X is separated/proper iff for all curves C and all maps f : C \ c -> X, f extends to C in at most/exactly one way.

Is there a good reason why this cannot possibly be true?

What I had in mind was something like the following:

X is separated/proper iff for all curves C and all maps f : C \ c -> X, f extends to C in at most/exactly one way.

Is there a good reason why this cannot possibly be true?

Here X denotes a reduced scheme of finite type of a field k (I guess people usually call this prevariety). I am mostly interested in the case where k is algebraically closed.