User:Addemf/sandbox/Lebesgue Differentiation (version 2.0)/Monotone Functions and Variation

Monotone Functions
While studying Fourier series, in 1829 the mathematician Dirichlet proved essentially the following theorem.

If a function is piecewise monotone and periodic, then it equals its Fourier series.

This perhaps established the initial interest in monotone functions.

A result sometimes shown in introductory analysis is that monotone functions cannot be discontinuous at more than a countable number of points. A natural question is whether monotonicity could give some similar bound on the points at which a function can fail to be differentiable.

It is not hard to find functions which are differentiable except on a set which is countable. Significantly harder is to construct a function is which is differentiable except on a set which is uncountable, but in fact such functions exist. In order not to go down a somewhat unnecessary rabbit hole, I will only refer the reader to the Cantor-Lebesgue function, which fails to be differentiable at all points on the Cantor set. The Cantor set, one may prove with some work, is uncountable but has measure zero.

All of this leads to the conjecture that

A monotonic function is differentiable almost everywhere.

Most of the remainder of this section is devoted to proving this result.

Dini Derivatives
While investigating these topics, the mathematician Dini realized that the usual characterization of the derivative gets in the way. In particular, there are too many possibilities for what the derivative may "be" at a point: It may exist, or it may be infinite, or it may fail to exist but not as an infinity.

Dini found a way to simplify matters, by defining the lower- and upper-derivatives. These are guaranteed to either exist or be infinity, as they are merely the limsup and liminf of the difference quotient (rather than the simple limit). Also, as we will see after stating their definition, they give us just as much information about the derivative as we need for our current purposes.

These will become important tools in the next lesson. Before that, we will want to introduce some more concepts and elementary results about them which will also prove important later.

Exercise 1. Prove the Lower/Upper Properties
Prove that the lower- and upper-left derivative for any function, always exists or is infinite. Also prove that the left-handed derivative exists if and only if the lower-left and upper-left derivatives are equal.

Also prove that if the function is monotonically increasing, then both the lower- and upper-derivative are nonnegative.

Partitioning Intervals
In 1881, the mathematician Camille Jordan understood that monotonicity was a sufficient condition for a function to equal its Fourier series. But he wondered:

Is this condition also necessary? That is to say, if a function equals its Fourier series then must it be piecewise monotone?

Let's not worry too much about the periodicity condition right now. There are fairly simple methods to relate a function on a bounded interval to a periodic function -- but those considerations are more suited for a course in Fourier analysis.

Thinking more about monotonicity, though, will lead to some important results.

If we think about the function $$f(x)=x^2$$, this is monotonic $$(-\infty,0]$$ and then again montonic on $$[0,\infty)$$.

Suppose that we wanted a Fourier series for this function on, say, the interval $$[-1,1]$$. We could approach this by first finding Fourier series for two different monotonic functions,


 * $$ f^-(x) = -\mathbf 1_{[-1,0)}x^2, \quad f^+(x)=\mathbf 1_{[0,1)}x^2 $$

This decomposition in fact makes both functions increasing, because I have taken the "decreasing part" and turned it upside-down. This choice is arbitrary, since we could also discuss monotonically decreasing functions. But it is just a bit simpler to just discuss monotonically increasing functions, and so we will prefer to do that.

Now notice that $$f = f^+-f^-$$. Although we don't want to become too distracted by working out the details of Fourier analysis, we should just appreciate that if we have a Fourier series for $$f^+$$ and for $$f^-$$, then by combining them, we should get a Fourier series for f.

Can Every Function Be Decomposed into Increasing and Decreasing Parts?
It is clear enough (for our purposes, anyway) that any function which can be decomposed into increasing and decreasing parts, has a Fourier series on any compact interval. This naturally provokes the question, "Can every function be decomposed into increasing and decreasing parts?"

If one thinks of the cosine function, on all of $$\Bbb R$$, then one could find its increasing part. The decomposition is more complicated than that for $$x^2$$. It would require not just using the indicator function once, but infinitely many times, to "turn off" the function when it switches from increasing to decreasing.

However, one can continue to think of more and more pathologically ill-behaved functions. As always, the Dirichlet function comes to mind -- can that be expressed as a difference of two monotonically increasing functions? It seems unlikely, but how could we prove it?

It makes sense that we extend our finite practice of partitioning the domain, into an infinite procedure.

In effect, each partition, P, becomes closer to a partition of f into increasing and decreasing parts. On each part, if f is increasing, then we hope that the positive variation captures the increase in f. And if f is decreasing on some part then we hope that the positive variation simply "ignores" these values. Thereby, we hope that the positive variation can account for the increasing part of f only.

In particular, we will want to let the end-point of the interval become a variable determined for each x. Then we will try to identify the increasing part of f by


 * $$f^+(x) = \mathcal V_f([a,x]) $$

and identify the decreasing part by


 * $$f^-(x) = \mathcal V_f([a,x]) - f(x)$$

with both being monotonically increasing functions, and $$f = f^+-f^-$$.

Here I want to emphasize my use of the word "try". We are not guaranteed that these objects exist at all. Even if they do, we are not yet guaranteed that they satisfy the properties stated above.

Exercise 2. Dirichlet Has Infinite Variation
Prove that if f is the Dirichlet function, then $$\mathcal V_f^+([0,1]) = \infty$$.

Exercise 3. Total Variation Is Increasing
Prove that for any function $$f:[a,b]\to\Bbb R$$, the associated total variation function $$T(x) = \mathcal V_f([a,x])$$ is a monotonically increasing function.