Let n∈ℕ, where the function f:A⊆ℝ𝑛→ℝ, 𝐴 is Borel, and (A,d) is a completely seperable space.

What is the name of the functions where the graph of f in any (n+1)-dimensional box of A×ℝ is countably infinite.

I know such a function is everywhere surjective (i.e., functions defined on a topological space whose restrictions to any non-empty open subset are surjective), but I don't know what type of everywhere surjective function it is.

For example there are strongly everywhere surjective functions and perfectly everywhere surjective functions. These definitions are thoroughly explained in the attatchment.

Do either accomplish what I seek?

Edit: I added the attatchment.