Is the world $C^\infty$?

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!)

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.

[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.