Regarding equations, for a physically unrelated but similar cognitive process, see[0] how we (A) set up a physical wave system, use complex numbers to work with wave equations, then liberally discard imaginary parts as if they never existed to get a physical result (B). (A) and (B) are anchored in physical reality, but the mathematical process in between is — as far as we know — not.

As for discontinuities, another non-thermodynamic physical discontinuity is the speed of light. Special relativity predicts it's entirely possible for particles to move FTL, when it's actually crossing the boundary that's a problem.

[0]: http://en.wikipedia.org/wiki/Plane_wave#Complex_exponential_...