To my knowledge, most autodiff tools will tend to silently ignore discontinuities, as long as you don't compute them at a non-differentiable point. E.g. if you

To my knowledge, most autodiff tools will tend to silently ignore discontinuities, as long as you don't compute them at a non-differentiable point. E.g. if you have y = abs(x) you will get back the derivative g = sign(x). This works as long as you don't try x=0. Similar things would happen for floors, rounding, if statements, etc. In general, as long as each local operations is differentiable, the whole program will be. That's isn't too hard to check.