My favourite example this is Mark Braverman's demonstration that all Quadratic Julia sets are computable (as in computable real numbers: meaning that you can produce arbitrary fine approximations).
However, as Mark notes, this proof proceeds by deriving 5 alternative programs to do the computation. For any given Julia set parameter 'c' (given as a computable real number) one of the 5 programs will compute the Julia set. Which one? Who knows. It depends discontinuously on the properties of 'c'.
However, as Mark notes, this proof proceeds by deriving 5 alternative programs to do the computation. For any given Julia set parameter 'c' (given as a computable real number) one of the 5 programs will compute the Julia set. Which one? Who knows. It depends discontinuously on the properties of 'c'.