# Converting extrapolation point to interpolation point

Just wondering if this is possible via mathematical conversion. I thought some Kernels might do this in the kernel space.

I got curious whether it's possible to create a general method that mathematically converts any collection of extrapolation points given some data into an interpolation points in the transformed space.

Mathematically speaking:

Does there always exist some function $$f:X \rightarrow Y$$ such that $$f(A) \subset f(U)$$ holds for any set of exterior points $$A \subset X$$ of some set of interpolation region $$U \subset X$$, $$U \cap A = \emptyset$$. If not, under which conditions would that hold?