Prove that for any two distinct points of an irreducible curve there exists a rational function that is regular at both, and takes the value 0 at one and 1 at the other.
I think I can construct such a function, for example, for given two points and . However, this doesn't work for general algebraically closed field, for example, the case of . Hence now I have no clue. Could you give me a hint for this problem?