Do regular functions on an algebraic variety separate points?

Define a variety as a reduced separated scheme of finite type over an algebraically closed field.

Given two closed points $$P$$ and $$Q$$ on a variety, does there exists an open set containing them and a regular function on it, such that one of $$f(P)$$ and $$f(Q)$$ is zero and the other non-zero?

The question came to mind when thinking about equivalent definitions of separatedness.

What I have tried so far: the answer is 'yes' if the points are contained in an open affine set. But there exist varieties such that some two points are not simultaneously contained in any open affine, as explained here.

Also, if the scheme is not separated, then it is not necessarily true, as shown by the example of a line with two origins.