I think we’ve developed a nice theme over the last few weeks, gradually working our way through a less well-behaved function – the triangle wave – and trying to find various series expansions for it. “Well-behaved” is kind of a term of art, which mathematicians use as shorthand for long strings of caveats about differentiability and continuity and which physicists use because who wants to memorize those long strings?
Well, to some extent we have to. While it’s not very common that badly-behaved functions arise in physics, there are functions which at least don’t always remember to say please and thank you. They have to be gently corrected, but they’re good at heart. The mathematicians are the ones who have to deal with the truly shady functions, the ones who form prison gangs and don’t play by the rules and obey the laws. Or theorems.
This one is in the latter category. It’s the ominous Blancmange function, a classic pathological case. It’s continuous everywhere and differentiable nowhere. There’s not a single point anywhere on its domain where it’s smooth, every single point is as prickly as a porcupine.
Via Wolfram, its graph:
As ludicrous as it is to have continuity everywhere without differentiability, it’s not actually hopelessly bad. While it’s true that continuity doesn’t imply differentiability, it is true that continuity implies integrability. The integral of this function over its domain [0,1] is just 1/2. Since it’s integrable, expansion in terms of the various orthogonal sets of functions that we’ve been doing will still work. We could pretty easily do a Fourier series or a series in Legendre polynomials. Or we could if we could do the necessary integrals, which I don’t think I’d want to try but could do numerically if worse came to worst.
What can you do with the Blancmange function? Heck if I know. If I ever got to teach a Calc 1 course I could use it as a good example of the fact that you can’t always take a statement to be mathematically certain just because it’s intuitively reasonable. While most commonly encountered continuous functions are differentiable, not all of ’em are. In fact by most set-theoretical standards such functions are actually the exception.
Fortunately they’re exceptions we don’t usually have to deal with. But like the outlaw mobsters of another era, they have their own rogue appeal.