Sunday, 23 December 2012

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



I am in search of a an analytic function f:RR 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

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...