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).