Abstract
On investigating the Weierstrass function: Its applications and impacts to the instruction in calculus Nadia Syeda, State University of New York, University at Buffalo The Weierstrass function is continuous everywhere but differentiable nowhere, is used as an example showing that continuity does not imply differentiability. The "deplorable evil" Weierstrass function ( Poincare,1893), had a revolutionizing impact in the mathematical community as it became an ideal counterexample breaking through the misconception that continuity entails differentiability. Unfortunately, most calculus students do not have much opportunity to investigate the function other than seeing it as an oddly behaved function. This project defies the norm, studying the function in threefold: (1) Research the historical motivation for the project, (2) investigate the non-differentiable behavior of the function, and (3) develop an instructional resource for college calculus in studying this specific function. The threefold approach to this study is unique because it serves as an example of how a mathematical study can thrust the envelope of venturing beyond the realm of formal proofs and analyses. Historical Motivation: Back in 1893 during the midst of the development of calculus ideas, there was a general belief that continuity of a function is sufficient, guaranteeing that the function is differentiable (Hermite's letter to Stieltjes, 1893). Weierstrass developed a function as a counter-example, a function with no derivatives, and it was labeled as "deplorable evil" (Poincaré, YEAR). He established a function that was differentiable at no point of location of the respective function yet was still continuous everywhere. Graphically speaking, Weierstrass function is a curve that has no gradient. To date, the function remains of interest to many, but rarely can it be seen in college calculus. The analysis portion begins with a review of the differentiability entails continuity, whereas the converse is false, such as f(x) = |x| showing that x=0 is continuous but not differentiable. Graphically, a point that is continuous but not differentiable is not "smooth," and this notion extends to the Weierstrass function where the curve is continuous everywhere, but "smooths" nowhere. This study reviews the established proof of non-differentiability, and begs the question, "What happens if one attempts to differentiate the Weierstrass function.
