continuous nowhere monotonic function

Let f be a real-valued continuous functionMathworldPlanetmathPlanetmath defined on the unit interval [0,1]. It seems intuitively clear that f should be monotonicPlanetmathPlanetmath on some subinterval I of [0,1]. Most of the concrete examples seem to supportMathworldPlanetmath this. A counterexample is termed nowhere monotonic, meaning that the function is not monotonic in any subinterval of [0,1]. Surprisingly, nowhere monotonic functions do exist:

Proposition 1.

There exists a real-valued continuous function defined on [0,1] that is nowhere monotonic.

A sketch of the proof goes as follows:

  1. 1.

    Let X be the set of all continuous real-valued functions on [0,1]. Then X is a complete metric space given the sup norm. Clearly X is non-empty.

  2. 2.

    Given any subinterval I of [0,1], the subset P(I)X consisting of all non-decreasing functions, the subset Q(I)X consisting of all non-increasing functions, and hence their union M(I), are closed.

  3. 3.

    Furthermore, M(I) is nowhere dense.

  4. 4.

    Let S be the set of all rational intervalsMathworldPlanetmathPlanetmath in [0,1] (a rational interval is an interval whose endpoints are rational numbers). Then S is countably infiniteMathworldPlanetmath. Take the union M of all M(I), where I ranges over S.

  5. 5.

    If f is monotone on some interval J in [0,1], then f is monotone on some rational interval IJ. If the theorem is false, then every continuous function is monotone on some rational interval, which means M=X.

  6. 6.

    However, M is a countableMathworldPlanetmath union of nowhere dense sets and X is a non-empty complete metric space. By Baire Category Theorem, this can not happen. Therefore, MX strictly and there exists a continuous nowhere monotone real-valued function defined on [0,1].

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 f, is given by a series


where the functions fk are defined by

f0(x)={x,if  0x12-x+1,if12x1  and  fk(x)=12kf0(2kx)

where each fk(x) is defined on [0,2-k]. Since fk agrees on the endpoints, we can extend the its domain to the entire unit interval by periodic extension (so that the graph of fk(x) has the shape of a sawtooth).

Figure 1: The graphics of f0 (left), f1 (middle) and f2 (right).

Figure 2: The graphic of the van der Waerden function (the dashed lines are the graphics of each fk).

- It is easy to check that each fk is continuous. Using the Weierstrass M-testMathworldPlanetmath we can also see that the series converges uniformly, and therefore conclude that f itself is a continuous function (it is the uniform limit of continuous functions).

- We now prove that f is nowhere monotonic:

The set {L2k:k, 0<L<2k} is dense in [0,1]. Given any interval I[0,1] we can then find a point of the form L2k in its interior.

It is easily seen that fj(L2k)=0 for jk.

For any integer j>k, consider the points aj:=2j-kL-12j and bj:=2j-kL+12j. The points aj (resp. bj) are just the points on the left (resp. on the right) of L2k when we divide the unit interval in segments of size 12j.

A direct calculation would show that


and similarly for bj.

Evaluating f in the points aj and bj we obtain

f(aj) = s=0fs(aj)
= s=0j-1fs(aj)
= s=0k-1fs(aj)+s=kj-1fs(aj)
= s=0k-1fs(L2k)+s=0k-1(±12j)+j-k2j
= f(L2k)+s=0k-1(±12j)+j-k2j

and similarly for f(bj).

The least value we can obtain is f(aj)=f(L2k)-k2j+j-k2j=f(L2k)+j-2k2j, and even in this extreme case we can still choose j large enough so that ajI and j>2k.

For this appropriate j we see that f(aj)>f(L2k), and similarly f(bj)>f(L2k).

Recall that aj<L2k<bj. We conclude that f is not monotonic in I, and hence it is nowhere monotonic.

Remark. The van der Waerden function turns out to be nowhere differentiableMathworldPlanetmathPlanetmath as well.

Title continuous nowhere monotonic function
Canonical name ContinuousNowhereMonotonicFunction
Date of creation 2013-03-22 14:59:08
Last modified on 2013-03-22 14:59:08
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 19
Author asteroid (17536)
Entry type Result
Classification msc 54E52
Defines van der Waerden function