The scientific thought that took shape during the Renaissance with Galileo and Descartes sought regularity in physical phenomena and, in parallel, in their mathematical models. Although the divine hypothesis was no longer the driving force behind our understanding of the world, belief in God remained strong, and there was no doubt that Mother Nature fulfilled His desire for beauty, a quality embodied in the regularity of the world. This may explain why the notion of discontinuity was entirely absent from scientific research.
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Dirichlet and trigonometric functions
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To discuss discontinuous functions, one must first define precisely what a function is—and, above all, what continuity means! Until the end of the 18th century, a function was defined by one expression, or sometimes two, composed of powers, roots, and logarithms and therefore de facto continuous (see les Fonctions, Bibliothèque Tangente 56, 2016). The effort to establish a precise notion of continuity began early in the following century with Bolzano and Cauchy and reached completion around 1860 with Karl Weierstrass and his students.
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