TITLE: Does there exist a smooth function which is nowhere analytic?
QUESTION [7 upvotes]: Smooth means has derivatives of all order, and analytic means can be given as a convergence of power series.

REPLY [5 votes]: A usual example is 
$$
f(x)=\sum_{k\in A} e^{-\sqrt{k}}\cos kx
$$
where $A=\{2^n:n\in\mathbb{N}\}$.