real analysis - Is there an analytic function which is not monotone on any interval?

I am in search of a an analytic function $f:\mathbb{R} \to \mathbb{R}$ which is not monotone on any nonempty open interval. Does one exist, or is there a proof that no such function exists?

If there does not exist such a function, is there an example of an infinitely differentiable function which is not monotone on any interval?

Answer

If $f$ is continuously differentiable, so in particular if it is twice differentiable, then $\{x:f'(x)>0\}$ and $\{x:f'(x)<0\}$ are open, and unless $f$ is constant at least one of the sets is nonempty. On an open interval in one of these sets, $f$ is monotone.

For differentiable functions that are not monotone in any interval, see the question "Differentiable+Not monotone."