The case most often cited as an example of a nondifferentiable function is derived from a sequence fn(x), each of which is a string of isosceles right triangles whose hypotenuses lie on the real axis and have length 1/n. As n→∞, the triangles shrink to zero size. For any finite n, the slope of fn(x) is ±1 almost everywhere. Then what happens as n→∞? The limit f∞(x) is often cited carelessly as a nondifferentiable function. Now it is clear that the limit of the derivative, fn′(x), does not exist; but it is the derivative of the limit that is in question here, f∞(x)≡0, and this is certainly differentiable. Any number of such sequences fn(x) with discontinuous slope on a finer and finer scale may be defined. The error of calling the resulting limit f∞(x) nondifferentiable, on the grounds that the limit of the derivative does not exist, is common in the literature. In many cases, the limit of such a sequence of bad functions is actually a well-behaved function (although awkwardly defined), and we have no reason to exclude it from our system.
Lebesgue defended himself against his critics thus: ‘If one wished always to limit himself to the consideration of well-behaved functions, it would be necessary to renounce the solution of many problems which were proposed long ago and in simple terms.’ The present writer is unable to cite any specific problem which was thus solved; but we can borrow Lebesgue’s argument to defend our own position.
To reject limits of sequences of good functions is to renounce the solution of many current real problems. Those limits can and do serve many useful purposes, which much current mathematical education and practice still tries to stamp out. Indeed, the refusal to admit delta-functions as legitimate mathematical objects has led mathematicians into error...
But the definition of a discontinuous function which is appropriate in analysis is our limit of a sequence of continuous functions. As we approach that limit, the derivative develops a higher and sharper spike. However close we are to that limit, the spike is part of the correct derivative of the function, and its contribution must be included in the exact integral...
It is astonishing that so few non-physicists have yet perceived this need to include delta-functions, but we think it only illustrates what we have observed independently; those who think of fundamentals in terms of set theory fail to see its limitations because they almost never get around to useful, substantive calculations.
So, bogus nondifferentiable functions are manufactured as limits of sequences of rows of tinier and tinier triangles, and this is accepted without complaint. Those who do this while looking askance at delta-functions are in the position of admitting limits of sequences of bad functions as legitimate mathematical objects, while refusing to admit limits of sequences of good functions! This seems to us a sick policy, for delta-functions serve many essential purposes in real, substantive calculations, but we are unable to conceive of any useful purpose that could be served by a nondifferentiable function. It seems that their only use is to provide trouble-makers with artificially contrived counter-examples to almost any sensible and useful mathematical statement one could make. Henri Poincaré (1909) noted this in his characteristically terse way:
In the old days when people invented a new function they had some useful purpose in mind: now they invent them deliberately just to invalidate our ancestors’ reasoning, and that is all they are ever going to get out of them.
We would point out that those trouble-makers did not, after all, invalidate our ancestors’ reasoning; their pathology appeared only because they adopted, surreptitiously, a different definition of the term ‘function’ than our ancestors used. Had this been pointed out, it would have been clear that there was no need to modify our ancestors’ conclusions...
Note, therefore, that we stamp out this plague too, simply by our defining the term ‘function’ in the way appropriate to our subject. The definition of a mathematical concept that is ‘appropriate’ to some field is the one that allows its theorems to have the greatest range of validity and useful applications, without the need for a long list of exceptions, special cases, and other anomalies. In our work the term ‘function’ includes good functions and well-behaved limits of sequences of good functions; but not nondifferentiable functions. We do not deny the existence of other definitions which do include nondifferentiable functions, any more than we deny the existence of fluorescent purple hair dye in England; in both cases, we simply have no use for them.
--E. T. Jaynes, Probability Theory (2003, pp. 669-71)
It’s somewhat incredible to read this while simultaneously picking up some set theory. It reminds me not to absorb what’s written in the high-status textbooks entirely uncritically, and to keep in mind that there’s a good amount of convention behind what’s in the books.
As Gauss stressed long ago, any kind of singular mathematics acquires a meaning only as a limiting form of some kind of well-behaved mathematics, and it is ambiguous until we specify exactly what limiting process we propose to use. In this sense, singular mathematics has necessarily a kind of anthropomorphic character; the question is not what is it, but rather how shall we define it so that it is in some way useful to us?
It’s somewhat incredible to read this while simultaneously picking up some set theory. It reminds me not to absorb what’s written in the high-status textbooks entirely uncritically, and to keep in mind that there’s a good amount of convention behind what’s in the books.