The Puzzling Case of the Abruptly Increasing Function

In the late 19th century, Karl Weierstrass invented a fractal-like function that was decried as nothing less than a “deplorable evil.” In time, it would transform the foundations of mathematics.

Calculus is a powerful mathematical tool. But for hundreds of years after its invention in the 17th century, it stood on a shaky foundation. Its core concepts were rooted in intuition and informal arguments, rather than precise, formal definitions. Two schools of thought emerged in response: French mathematicians who were content to apply calculus to problems in physics, and German mathematicians who sought to tear things down by finding counterexamples that would undermine long-held assumptions.

One of these mathematicians was Karl Weierstrass. Though he showed an early aptitude for math, his father pressured him to study public finance and administration, with an eye toward joining the Prussian civil service. Bored with his university coursework, Weierstrass is said to have spent most of his time drinking and fencing; in the late 1830s, after failing to get his degree, he became a secondary school teacher, giving lessons in everything from math and physics to penmanship and gymnastics.

Weierstrass didn’t begin his career as a professional mathematician until he was nearly 40. But he would go on to transform the field by introducing a mathematical monster.

In 1872, Weierstrass published a function that threatened everything mathematicians thought they understood about calculus. He was met with indifference, anger, and fear, particularly from the mathematical giants of the French school of thought. Henri Poincaré condemned Weierstrass’ function as “an outrage against common sense.” Charles Hermite called it a “deplorable evil.”

To understand why Weierstrass’ result was so unnerving, it helps to first understand two of the most fundamental concepts in calculus: continuity and differentiability. A continuous function is exactly what it sounds like—a function that has no gaps or jumps. You can trace a path from any point on such a function to any other without lifting your pencil.

Calculus is in large part about determining how quickly such continuous functions change. It works, loosely speaking, by approximating a given function with straight, nonvertical lines. At any given point on this curve, you can draw a “tangent” line—a line that best approximates the curve near that point. The slope, or steepness, of the tangent line measures how quickly the function is changing at that point.

Weierstrass discovered a function that, according to Ampère’s proof, should have been impossible: It was continuous everywhere yet differentiable nowhere. He built it by adding together infinitely many wavelike “cosine” functions. The more terms he added, the more his function zigzagged—until ultimately, it changed direction abruptly at every point, resembling an infinitely jagged sawtooth comb.

The Weierstrass function has no derivative at any point, yet it is continuous and even uniformly continuous on the entire real line.

This paradoxical property makes it a fascinating example in mathematical analysis.

” tern=”weierstrass function” ]

Many mathematicians dismissed the function. It was an anomaly, they said—the work of a pedant, mathematically useless. They couldn’t even visualize it. At first, when you try to plot the graph of Weierstrass’ function, it looks smooth in certain regions. Only by zooming in will you see that those regions are jagged, too, and that they’ll continue to get more serrated and badly behaved with each additional magnification.

But Weierstrass had proved beyond doubt that, though his function had no discontinuities, it was never differentiable. To show this, he first revisited the definitions of “continuity” and “differentiability” that had been formulated decades earlier by the mathematicians Augustin-Louis Cauchy and Bernard Bolzano. These definitions relied on vague, plain-language descriptions and inconsistent notation, making them easy to misinterpret.

Karl Weierstrass didn’t begin his mathematical career until he was nearly 40. His dedication to rigor and logic ultimately led to the birth of modern analysis.

The proof demonstrated that calculus could no longer rely on geometric intuition, as its inventors had done. It ushered in a new standard for the subject, one that was rooted in the careful analysis of equations. Mathematicians were forced to follow in Weierstrass’ footsteps, further sharpening their definition of functions, their understanding of the relationship between continuity and differentiability, and their methods for computing derivatives and integrals.

Today, we know that mathematics is full of monsters: impossible-seeming functions, strange objects (it’s one of the earliest examples of a fractal), wild behaviors. “There’s a whole universe of possibilities, and the Weierstrass function is supposed to be opening your eyes to it,” said Philip Gressman of the University of Pennsylvania.

It also turned out to have many practical applications. In the early 20th century, physicists wanted to study Brownian motion, the random movement of particles in a liquid or gas. Because this movement is continuous but not smooth—characterized by rapid and infinitely tiny fluctuations—functions like Weierstrass’ were perfect for modeling it. Similarly, such functions have been used to model uncertainty in how people make decisions and take risks, as well as the complicated behavior of financial markets.

Much like Weierstrass himself, the consequences of his function have sometimes been late to bloom. But they’re continuing to shape mathematics and its applications today.