I am not a historian of mathematics, but reading Penrose’s book The road to reality I understood a conceptual point that deserves attention. At the beginning of nineteenth century Fourier proved that an infinite sum ( a series) of harmonic functions could give rise to a non continuous periodic function, that is to a square wave. Here you can see practically how this works. Surely this was not sufficient to cause the passage from an algebraic representation of the notion of function to an insiemistic one. Penrose speaks of a well-educated function. In nineteen mathematics well-educated function were polinomial and harmonic functions. Now a function is a set of couples from a domain to a range with the constrain that not two members of the range could be associated to the same memeber of the domain. It seems that the proof that infinite sums of well-eductaed functions in the old sense result in a non well-educated function, opened the way to an enlargemet of the concept of politeness of functions.