Derivation or antiderivation, when  you  hold  us…

Jean-Baptiste Hiriart-Urruty Institut de math´ematiques Universit´e Paul Sabatier

118, route de Narbonne

31062 Toulouse Cedex 9, France www.math.univ-toulouse.fr/ ˜jbhu/

“To take the derivative,  to take the antiderivative  (or primitive), to integrate func- tions…”, all these expressions are familiar to the students in sciences, especially in math-

ematics. We revisit here some of their aspects, comparing essentially three classes of func- tions: the class C(I ) of continuous functions on the interval I , the class D(I ) of functions

 which are derivatives (or from which we can construct the so-called antiderivatives), the class VI(I ) of functions satisfying the intermediate value property.  The string of inclusions

between these sets is:

C(I ) ⊂ D(I ) ⊂ VI(I ).

The main objective of the present work is to comment these inclusions, more specifically the gaps between them, and to consider the effect of multiplying two derivative functions. Here is the plan:

1. The inclusion C(I ) ⊂ D(I ): Every continuous function is a derivative function.

2. The inclusion D(I ) ⊂ VI(I ): Theorem of G.Darboux.

3. The class D(I ) is not stable by multiplication: the product of two derivative functions

is not always a derivative function.

4. Final observations and possible extensions.

Prerequisites: Calculus, such as taught in the first two years of universities.

A paper written in French, entitled “D´erivation ou primitivation,  quand tu nous tiens…”,

to be published in the Bulletin of teachers in mathematics (Bulletin de l’APMEP), will be

available at the end of the talk.

References

[1] V.Cercle´, Fonctions sans primitive.  Bulletin de l’APMEP n◦505, pages 427 − 434

(2013).

[2]  S.B.Nadler, A proof of Darboux theorem.   American Math.   Monthly 117, pages

174-175 (2010).

[3]  J.-B.Hiriart-Urruty,  Que manque-t-il a` une fonction  v´erifiant la propri´et´e des valeurs interm´ediaires pour ˆetre continue ? Revue de Math´ematiques  Sp´eciales  94, pages

370 − 371 (1984).

[4]  W.Wilcosz, Some properties of derivative functions. Fundamenta MathemaI, pages 145 − 154 (1921)