Is the world $C^\infty$? 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).
 A: I am not even sure if the world is $C^{0}$. The concept of uncountability in "real" world is still hard for me to digest. I am happy to deal with uncountability in pure mathematics but I am not sure if it is the case in the "real" world. It might be possible to reformulate all of physics in terms of discrete and not continuous.
One such attempt is Discrete Philosophy though I don't know how much of this is true and how much is not. See Digital Philosophy
It might be possible to reformulate them in terms of some fundamental quantities and assume that these quantities cannot be subdivided further. For instance, discretize space in terms of say Planck's length and time in terms of say Planck's time and so on.
A: [Some very nice answers by Eric, Sivaram and Piotr above. Here's my take!]
Short answer: NO !
The notion of $C^\infty$ is a mathematical aberration that was conjured up to help smooth (pun intended) discussions in real analysis.
Now, remember, you asked "Is the world $C^\infty$?". By "world" I take it to mean the physical world around us, our notions of which are based on what we can observe. A physical observable which is infinitely differentiable, would require an infinite number of measurements to determine the value of that observable in a given region.
Given that the consensus is emerging that information is the underlying substrate of the Universe (in the various forms of the holographic principle), it becomes even more urgent to reject a notion of $C^\infty$ observables.
Note how I have stressed the words "physical observables" rather than functions or mathematical entities that are used as intermediaries to compute any measured quantity. This is in harmony with Eric's statement that:

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.

A: What quantities supposed to be $C^\infty$? 
I don't know if it answers you question, but AFAIK smooth functions are nice and useful tool to describe many aspects of the physical world. However, I don't see why they should be considered as fundamental in any sense.
When it comes to QFT, even there you often encounter Dirac delta (and you can't get rid of it easily).
One professor from my department when asked if all physical dependences are continuous answered "Yes - and even more - with discrete domain" (as you will never make an infinite number of measurements).
Anyway, in my opinion there may be more specific (and purposeful) questions:


*

*If for a given theory such-and-such dependences are continuous/analytic/smooth/(other nice property)?

*If in practice one can restrict oneself to using only smooth functions, resulting in approximation error below measurement error?

A: 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!)
A: Another counter-example would be the voltage fluctuations across a resistor due to thermal noise. This is a white-noise which is continuous everywhere, but not derivable at any point.
