Monster Function that Broke Calculus

Calculus, independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, has powered our understanding of science and driven technological progress from the industrial revolution to modern artificial intelligence.

In this episode, we explore the jagged function that broke calculus, examining how mathematicians for two centuries relied on the intuitive dogma that continuous, unbroken functions must be smooth and differentiable almost everywhere, as claimed in André-Marie Ampère's 1806 proof.

We examine how Karl Weierstrass shattered this comfortable world in 1872 by presenting a function constructed from an infinite series of cosine waves that is continuous everywhere yet differentiable nowhere.

Finally, we investigate the profound shock this "deplorable evil" caused among contemporaries like Henri Poincaré and Charles Hermite, forcing mathematicians to abandon flawed intuition and rebuild mathematical analysis from the ground up.