How many smooth functions are non-analytic? 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)?

In response to your first question, it's not hard to see there are continuum many continuous functions, and hence continuum many analytic functions (consider the values of said functions at rational points; also, the constant functions provide at least continuum many.
On the other hand, there are 2^c many Lebesgue measurable functions of the reals: If S is any subset of the cantor set, the indicator function of S is Lebesgue measurable, since it has measure 0. There are, however, only continuum many Borel measurable functions.
Maybe you meant to ask, "How many smooth functions are not analytic?" I can sort of see a picture proof in my had that in the topology of continuous functions of the reals, non-analytic functions are a dense subset of the set of smooth functions.
