Abstract / Synopsis
At the turn of the 19th century, mathematics developed the rigor that is now considered essential to the field. Mathematicians began going back and proving theorems and statements that had been taken to be true on face value, ensuring the underpinnings of mathematics were solid. This era of mathematics was characterized not only by setting foundations, but also by pushing the boundaries of new ideas. In 1830, Bolzano found an example of a function that was nowhere differentiable, despite being continuous. Thirty years later, Cellerier and Riemann each discovered another example of such a pathological function. The first everywhere continuous nowhere differentiable function to be published appeared in Borchardt's Journal in 1875. This function was proposed by Karl Weierstrass. His function, in addition to similar examples that followed it, revolutionized the ideas of continuity, differentiability, and limits. Due to these pathological functions, the traditional definitions were rethought and revised. Here we explore some of these pathological functions.
Shoemaker, A. "On French Pudding and a German Mathematician," Journal of Humanistic Mathematics, Volume 7 Issue 2 (July 2017), pages 239-258. DOI: 10.5642/jhummath.201702.11 . Available at: http://scholarship.claremont.edu/jhm/vol7/iss2/11