Analyticity of infinitely oscillatory functions I have a simple question: if $f:[-1,1]\to\mathbb{R}$ is analytic, can it be infinitely oscillatory (by which I mean something like $\sin(1/x)$ ?

No, if $U=f^{-1} (f(x))$ is the preimage of $f(x)$ then $x$ must be isolated in $U$.