We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather contrived, and most smooth functions that I've encountered in math and physics are analytic.
How many smooth functions are not analytic (in terms of measure or cardinality)? In what situations are such functions encountered? Are they ever encountered outside of real analysis (e.g. in physics)?
Best Answer
It is not difficult to see that the collection of $C^{\infty}$ functions that fail to be analytic at each point is $c$-dense in the space of continuous functions defined on a compact interval (sup metric). To see this, let $f$ be such a continuous function and choose $\epsilon > 0.$ Next, pick any $C^{\infty}$ and nowhere analytic function $\phi$ that is bounded between $-1$ and $1.$ (Take $\frac{2}{\pi}$ times the arctangent of an unbounded example, if an example bounded between $-1$ and $1$ isn't handy.) Let $P$ be a polynomial whose sup-metric distance from $f$ is less than $\frac{\epsilon}{3}$ (Weierstrass's Approximation Theorem). Now let $g = \left(\frac{\epsilon}{3}\right)\phi + P.$ Then, for each of the $c$-many real numbers $\delta$ such that $0 < \delta < \frac{\epsilon}{3},$ the function $g + \delta$ is: (a) $C^{\infty}$, (b) nowhere analytic, (c) belongs to the $\epsilon$-ball centered at $f.$ This last part involves the triangle inequality, and earlier we need the fact that if $\phi$ is $C^{\infty}$ and nowhere analytic, then the composition $\arctan \circ \phi$ is $C^{\infty}$ and nowhere analytic and $\left(\frac{\epsilon}{3}\right)\phi + P + \delta$ is $C^{\infty}$ and nowhere analytic. Note that we can also easily get $c$-many such functions arbitrarily close (sup metric) to any continuous function defined on $\mathbb R$ by appropriately splicing together functions on the intervals $...\; [-2,-1],$ $[-1,0],$ $[0,1],$ $[1,2],\; ...$
Any type of cardinality result is pretty much maxed out by this result, but by considering stronger forms of "largeness" we can do better. The results I know about involve Baire category and the idea of prevalance (complement of a Haar null set), and each implies the $c$-dense result above (and much more). Back in 2002 I posted a couple of lengthy essays in sci.math about $C^{\infty}$ and nowhere analytic functions. For some reason they were never archived by google's sci.math site, but they can be found at the Math Forum sci.math site. One day I might LaTeX these essays for posting in this group, but I doubt I'll have time in the near future.
ESSAY ON NOWHERE ANALYTIC C-INFINITY FUNCTIONS Part 1 (9 May 2002) and Part 2 (19 May 2002)