A few days ago, I posted about a conversation I had with my friend L. We spent some time catching up and, in so doing, spent a little time talking about this particular plot of space on the grand ol’ internet. He mentioned a couple blog topics for me to consider and also asked if I was contemplating research in algebra/topology; looking back, the fact that L’s an analyst, the fact that I have very few analysis posts here, and the fact that the topics he suggested were analysis topics made me realize I really do need to do a better job representing my enjoyment for analysis. Consider this entry step one of that, perhaps.
Rather than spending a bunch of time researching stuff I’d never seen before, I decided to type up a little summary thing of an interesting article I found online when I was a master’s student. For a little perspective as to why this particular article is important, we’ll have to take a trip into so-called higher education and examine the topic that generally serves as most people’s introduction to grown up mathematics, i.e., calculus. A (really really over-simplified, primitive, simplistic) synopsis of calculus can be summed up in this way: Calculus is a class that abstracts the unknown variable quantities thrown at you in Algebra I/II into unknown variable quantities that themselves can vary by way of limiting arguments.
And that’s basically it: In America, calculus is really just taught as the algebra of limits. As such, some basic limit-intrinsic notions such as continuity, differentiability, and integrability are touched on / hinted at, and at the end of fourteen weeks of being fooled into thinking you’re finally understanding what math is, you’re sent on your way. For most, that’s the end of the story, but for a self-selecting few, the journey through mathematics continues, and new techniques / ideas get thrown at you in hopes that they’ll stick and that you’ll be able to use them for something special….
…and at the same time, for that self-selecting few, it’s not uncommon at all for certain somewhat obvious questions to go unasked through the years. For example: It’s invariably shown in Calculus I that fails to be differentiable at because of the sharp edge there. It stands to reason, then, that combinations and scalings of the absolute value function with two, three, four, etc. sharp edges would fail to be differentiable at two, three, four, etc. values of . This idea isn’t a hard one to grasp for a calculus student. But then the next question: How many points of non-differentiability can a function have? Or how about, Construct a function that fails to be differentiable at infinitely many points. Most students would be quick to adapt previous examples and notice that a saw-blade function with sharp points at each value , , proves the existence of functions with infinitely many points of discontinuity. Again, no big deal.
So what, then? Can we have functions that are non-differentiable at uncountably many points? How about functions that are differentiable nowhere? By and large, these are ideas that escape lots of students – even students nearing the end of a traditional math major curriculum at an average American institution. I know this because I was once one of those students, and have since taught several myself: I see how students fail to comprehend non-differentiability and even the everywhere-discontinuity of functions like . It’s simply something that fails to register for the average student.
Coincidentally, it doesn’t always stop there. L was actually telling me a story once about a statistics professor we both knew who claimed, absent-mindedly, that most continuous functions are differentiable. That, of course, is a big statement, and for the inquisitive audience-member, the natural response is: Prove it. Hence the aforementioned paper….
As it turns out, this professor was wrong: In a very precise sense, most continuous functions are in fact nowhere differentiable, something that seems almost mind-boggling at first consideration but something that is, in any case, provable. To do so, we’ll need some background.
We’re going to let be a compact subset of and write for the space of all continuous real-valued functions defined on . For completeness, note that the norm of a function in is given by . In order to talk about most or almost none, there need to exist some notions of smallness of subsets of . To that end, we say that a subset is dense in if for all . Similarly, a subset of is said to be nowhere dense if for all open balls , there exists an open ball for which . Finally, the subset of is said to be meager if where the sets are all nowhere dense. In lots of literature, meager sets are called sets of first category, and sets which fail to be meager are said to be sets of second category. One result we’ll need which is stated without proof is attributed to Baire.
The Baire Category Theorem for .
Using the terminology above, is second category (i.e., not meager).
A more precise statement of the claim that “most” continuous functions are nowhere differentiable using the language of Baire is the purposes of this exposition, given here as a Theorem. First, let denote the set of functions in which are differentiable at some .
Theorem 1. is meager in , from which it follows that some – and indeed most – -functions have derivatives at no point in .
Before proceeding, the following technical lemma is stated. The proof is deferred for brevity.
Lemma 2. For a closed set in , is nowhere dense if and only if contains no open balls.
Lemma 2 makes sense, intuitively: A set that’s nowhere dense can be loosely thought of as a set with no meat to it, and because open balls (and indeed, open sets in most topologies) are thought of as sets with substance – indeed, open sets contain other open sets – it would make sense that sets without substance fail to contain sets with substance.
Proof of Theorem 1.
For each , define the set as follows:
Clearly, a function is in for some positive integers $n,m$ by definition of the derivative as a limit of difference quotients. This tells us that . Also, note that is closed and that contains no open balls, which by Lemma 2 says that is a nowhere dense set that contains Therefore,
is a meager set (as it’s the countable union of sets which are nowhere dense) that contains , thereby proving that itself is meager in . By the Baire Category Theorem for , however, it’s known that isn’t meager, and so it follows that “most” of lies outside of and hence is nowhere differentiable.
1. To prove this, let be a Cauchy sequence in for which . Note, then, that for each , there exists a point for which satisfies the condition defining . By compactness of , the sequence is a bounded sequence, which by the Bolzano-Weierstrass Theorem has a convergent subsequence ; hence, by passing to a subsequence, it may be assumed that converges to a point . Hence, when , it follows that
and so .
2. Let and consider for . It suffices to find a function which is in but not in . The result stems from the density of the collection of piecewise linear continuous functions in and from the freedom to choose such functions in a way that puts them “near” the functions and while leaving their slopes larger than the prescribed bounds allowed by . This construction is given in its entirety in the document linked both above and here for convenience.