continuous nowhere monotonic function
Let be a real-valued continuous function defined on the unit interval . It seems intuitively clear that should be monotonic on some subinterval of . Most of the concrete examples seem to support this. A counterexample is termed nowhere monotonic, meaning that the function is not monotonic in any subinterval of . Surprisingly, nowhere monotonic functions do exist:
There exists a real-valued continuous function defined on that is nowhere monotonic.
A sketch of the proof goes as follows:
Given any subinterval of , the subset consisting of all non-decreasing functions, the subset consisting of all non-increasing functions, and hence their union , are closed.
Furthermore, is nowhere dense.
Example : van der Waerden function
The above shows the existence of such a function. Here is an actual example of a nowhere monotonic continuous function, called the van der Waerden function. This function, which we designate by , is given by a series
where the functions are defined by
where each is defined on . Since agrees on the endpoints, we can extend the its domain to the entire unit interval by periodic extension (so that the graph of has the shape of a sawtooth).
- It is easy to check that each is continuous. Using the Weierstrass M-test we can also see that the series converges uniformly, and therefore conclude that itself is a continuous function (it is the uniform limit of continuous functions).
- We now prove that is nowhere monotonic:
The set is dense in . Given any interval we can then find a point of the form in its interior.
It is easily seen that for .
For any integer , consider the points and . The points (resp. ) are just the points on the left (resp. on the right) of when we divide the unit interval in segments of size .
A direct calculation would show that
and similarly for .
Evaluating in the points and we obtain
and similarly for .
The least value we can obtain is , and even in this extreme case we can still choose large enough so that and .
For this appropriate we see that , and similarly .
Recall that . We conclude that is not monotonic in , and hence it is nowhere monotonic.
Remark. The van der Waerden function turns out to be nowhere differentiable as well.
|Title||continuous nowhere monotonic function|
|Date of creation||2013-03-22 14:59:08|
|Last modified on||2013-03-22 14:59:08|
|Last modified by||asteroid (17536)|
|Defines||van der Waerden function|