While it is quite common to use piecewise constant functions to describe reality, e.g. the optical properties of a layered system, or the Fermi–Dirac statistics at (the impossible to reach exactly) $T=0$, I wonder if in a fundamental theory such as QFT some statement on the analyticity of the fields can be made/assumed/proven/refuted?
Take for example the Klein-Gordon equation. Even if you start with the non-analytical Delta distribution, after infinitesimal time the field will smooth out to an analytical function. (Yeah I know, that is one of the problems of relativistic quantum mechanics and why QFT is "truer", but intuitively I don't assume path integrals to behave otherwise but smoothing, too).
Answer
This is a really interesting, but equally beguiling, question. Shock waves are discontinuities that develop in solutions of the wave equation. Phase transitions (of various kinds) are non-continuities in thermodynamics, but as thermodynamics is a study of aggregate quantitites, one might argue that the microscopic system is still continuous. However, the Higgs mechanism is an analogue in quantum field theory, where continuity is a bit harder to see. It is likely that smoothness is simply a convenience of our mathematical models (as was mentioned above). It is also possible that smooth spacetime is some aggregate/thermodynamic approximation of discrete microstates of spacetime -- but our model of that discrete system will probably be described by the mathematics of continuous functions.
(p.s.: Nonanalyticity is somehow akin to free will: our future is not determined by all time-derivatives of our past!)
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