TITLE: Is there a version of the valuative criteria for separateness/properness for varieties?
QUESTION [11 upvotes]: 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.

REPLY [5 votes]: I would like to answer my own question after nearly 12 years.  I am teaching algebraic geometry this year and in the process I found the statement that I wanted as Corollary 15.10 in Gortz-Wedhorn (an excellent book).
Theorem
Let $Y $ be a locally Noetherian scheme and let $ f : X \rightarrow Y $ be a morphism of finite type.  Then $ f $ is proper if and only if for all normal absolute curves $C $ which are schemes over $Y $, for every closed point $ c \in C$, and for every $ Y $-morphism $ g : C \setminus \{c \} \rightarrow X $, there exists a morphism $ \bar g : C \rightarrow X $ such that $ \bar g|_{C \setminus \{c\}} = g $.
When I asked my question above, I was wondering if there was a more geometric version of the valuative criterion and that is provided by this theorem.  In the special case where $ Y = Spec k $, then $ C $ will be a regular curve over $ k $.