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…. Continue reading →

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