Disproving that a plane is the union of countably many functions. Where is the fallacy?

Question

This question is inspired by the following statement: "The plane is the union of countably many functions y=f(x) iff Continuum Hypothesis holds."

At first glance it is easy to show that a plane is NOT a union of countably many functions y=f(x).

Proof: Lets study line x=0. Each function crosses x=0 in exactly one point. If we can cover x = 0 with countably many functions then we can as well enumerate all real numbers with countable many points, which contradicts Cantor's first uncountability theorem. Hence, there is at least one point that isn't covered. QED (?)

Contor would have published it, if it was this easy. What went wrong?

Answer 1

We must consider two sets of functions: y=f(x) and x=g(y). It is easy to prove that one set of functions wouldn't cover the plane. It is unclear whether two sets of functions can cover the plane.