Introduction

From 2002 to 2006, the “history of mathematics” group of Paris 7 IREM contributed to a research project funded by the Institut National de la Recherche Pédagogique (INRP). We chose to work on the multiplicity of viewpoints on functions. In spite of the fact that some didactical and some historical research work was available on this topic, we felt the relevant connections still needed to be pointed to and explored. We also made use of fresh historical research work, namely R. Chorlay's doctoral dissertation on the emergence of the concepts of “local” and “global” in mathematics [7, 8].

We borrowed from didactical works the notion of viewpoint (as opposed to theoretical frame and semiotic register [12] and the distinction between four viewpoints in mathematical Analysis: point-wise, infinitesimal, local and global. Didactical work focused either on issues of cognitive flexibility (versatility)—the ability to change viewpoints, frames or semiotic registers in problem-solving—and its growing importance in higher education (Advanced Mathematical Thinking), or on curricular discontinuities: a point-wise/global dialectic when the concept of function is first encountered, then the infinitesimal and local viewpoints come into play with calculus, then an all-encompassing abstract theoretical frame in higher education. We focused our more historical investigation on a series of hot spots in which the four viewpoints interact and, eventually, were made explicit in the 19th century: proofs of the mean value theorem; proofs of the link between the sign of f′ and the variations of f; differentiation of the three notions of maximum, local maximum and upper bound; emergence of the domain concept.