Tangency vis-a'-vis differentiability by Peano, Severi and Guareschi

Details Tangency vis-a'-vis differentiability by Peano, Severi and Guareschi Tangency vis-a'-vis differentiability by Peano, Severi and Guareschi

2010-03-05

arxiv.org/abs/1003.1332v1

Mathematics

History and Overview

35 pp., on the occasion of the 150th anniversary of the birth of Giuseppe Peano (1858-1932); to appear in J. Convex Analysis

Scientific paper

Peano defined 'differentiability' of functions and 'lower tangent cones' in 1887, and 'upper tangent cones' in 1903, but uses the latter concept already in 1887 without giving a formal definition. Both cones were defined for arbitrary sets, as certain limits of appropriate homothetic relations. Around 1930 Severi and Guareschi, in a series of mutually fecundating individual papers, characterized differentiability in terms of 'lower tangent cones' and strict differentiability in terms of 'lower paratangent cones', a notion introduced, independently, by Severi and Bouligand in 1928. Severi and Guareschi graduated about 1900 from the University of Turin, where Peano taught till his demise in 1932.

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