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.