Are there any true discontinuities in physics?

Any process that causes a physical quantity to become truly discontinuous in space and/or time by definition takes place over an extremely (in fact, infinitely) short time or length scale. From the usual uncertainty principles of quantum mechanics, these process would have huge energy or momentum, and would presumably result in both very strong quantum and gravitational effects. Since we don't have a good theory of quantum gravity, there's really very little we can say with confidence about such extreme regimes.

But even if we do one day come up with a perfectly well-defined and self-consistent theory that reconciles quantum field theory with general relativity and is completely continuous in every way, that still won't settle your question. Such a theory can never be proven to be "the final theory," because there will always be the possibility that new experimental data will require it to be generalized. The most likely place for this "new physics" would probably be at whatever energy scales are beyond our current experimental reach at the time. So we'll probably always be the least confident of the physics at the very smallest of length or time scales.

A similar line of thought holds for the possibility of absolute discontinuities in energy or momentum: ruling out, say, really tiny discontinuities in energy would require knowing the energy to extremely high precision. But by the energy-time uncertainty relation, establishing the energy to such high precision would require an extremely long time - and eventually the required time scale would become too long to be experimentally feasible.

So extremely long and extremely short time/length scales both present fundamental difficulties in different ways, and your question will probably never be answerable.