When we first learn physics, it's often presented very 'discontinuously'. For example, pop quantum likes to talk about objects being "either" particles or waves, leading to a lot of confused questions about how things switch between the two. Once you learn about wavefunctions, the problem goes away; 'particle' and 'wave' are just descriptions of two extreme kinds of wavefunctions.
In general, further learning 'fills in' the knowledge holes that discontinuities cover up:
- Phase transitions in thermodynamics. These are only truly discontinuous in the $N \to \infty$ limit, which doesn't physically exist. For large but finite $N$, we can use statistical mechanics to get a perfectly continuous answer.
- Measurement in quantum mechanics. 'Copenhagen collapse' is not instantaneous, it's the result of interaction with an external system, which occurs in continuous time.
- Optical decays. Without QED, the best model is to just have atoms suddenly and randomly emit photons with some lifetime. With QED, we have a perfectly continuous time evolution (allowing for, e.g. Rabi oscillations).
At this point I'm having trouble thinking of any 'real' discontinuities. Are there any theories (that we believe to be fundamental) that predict a discontinuity in a physically observable quantity?
To address several comments: I am not looking for a discontinuity in time, as this is associated with infinite energy. I am not looking for experimental confirmation of a discontinuity in time, since that's impossible.
I am asking if there is any measurable parameter in any of our currently most fundamental theories which changes discontinuously as a function of another measurable parameter, according to the theory itself. For example, if phase transitions actually existed, then phase as a function of temperature or pressure would work.