I am in search of a an analytic function f:R→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."
No comments:
Post a Comment