Darboux Functions

A real valued function f defined on an interval I is said to nave the intermediate value property if whenever a and b are in I, and y is any number between f (a) and f(b), there is a number x between a and b such that f(x) = y. This property was believed, by some 19th century mathematicians, to be equivalent to the propertyof continuity. In 1875, Darboux showed that this belief was not justified. He proved that every derivative has the intermediate value property and he gave examples of some rather badly discontinuous derivatives. (We now know that a function having the intermediate value property can be discontinuous everywhere, and, in fact, be nonmeasurable on every set having positive measure.) Because of Darboux's work on the subject, one now usually calls a function having the intermediate value property a Darboux function.

Now, every derivative has the Darboux property. The same is true of every function belonging to any of several classes of functions which are related to the class of derivatives.

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Invention of the devil...? No.

Abel wrote in 1828 : "Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever".
In the ensuing period of critical revision they were simply rejected. Then came a time when it was found that something after all could be done about them. This is now a matter of course, but in the early years of the past century the subject, while in no way mystical or unrigorous, was regarded as sensational, and about the term "divergent series", now colourless, there hung an aroma of paradox and audacity.

Il teorema di Bolzano (1817) o di Weierstrass (1885)?

Theorem. If a function Fx is continuous from x=a to x=b inclusive, then among all the values which it takes, if we imagine that x successively takes all the values from a to b inclusive, there is always a greatest in the sense that no other is greater than it, and there is also a smallest in the sense that no other is smaller than it.

Dai manoscritti di Bernard Bolzano (Functionenlehre e Größenlehre 1817)

Paradoxes of the Infinite Dr. Bernard Bolzano 1851

Je suis tellement pour l’infini actuel, qu’au lieu d’admettre, que la nature l’abhorre, comme l’on dit vulgairement, je tiens qu’elle l’affecte par-tout, pour mieux marquer les perfections de son Auteur.

— Leibniz, Opera omnia studio Ludov. Dutens., Tom. II, part x, p. 243 

Certainly most of the paradoxical assertions which we meet with in the area of mathematics, though not all of them as Kästner suggests, are propositions that either contain the concept of infinity directly, or depend on it in some way for their attempted proof. It is even more indisputable that precisely those mathem- atical paradoxes which deserve our greatest attention are of this kind. This is because decisions on very important questions in many another subject, such as metaphysics and physics, depend on a satisfactory resolution of their apparent contradictions.

This is the reason why in the present work I am dealing exclusively with the consideration of the paradoxes of the infinite. But it is self-evident that it would not be possible to recognize the appearance of contradiction which is attached to these mathematical paradoxes for what it is, a mere appearance, if we did not make abundantly clear what concept we actually associate with the infinite. Therefore we do this first.