TITLE: Examples of statements that are true for real analytic functions but false for smooth functions
QUESTION [10 upvotes]: I'm writing because I don't know the usefulness of real analytic functions. I mean, I know that analyticity is something more respect differentiable ($C^\infty$ function), but I don't have in mind a result, which is true only for real analytic functions, and then become false for $C^\infty$ functions which are not analytic.

REPLY [5 votes]: The Cauchy-Kowalevski Theorem, together with Lewy's example, provides some food for thought.
The CKT is the main result about local existence of a solution of an analytic first order system of PDEs. Roughly speaking, if all the coefficients, the force and the boundary datum are analytic at some point the PDE admits a local solution which is $C^{\infty}$ and also analytic at the given point.
On the other hand, Lewy's example shows that the analogous theorem for smooth functions does not hold: he considered a system of PDEs with polynomial coefficients and was able to prove that there is a $C^{\infty}$ force for which the system has no solution of class $C^2$ on any open set. (Clearly this $f$ cannot be analytic in view of the CKT)
Needless to say, the proof of the CKT (at least the one I am aware of) is very involved and relies heavily on the power series expansion of the analytic functions.