The original version of this story appeared in Quanta Magazine. Calculus is a powerful mathematical tool. But for hundreds of years after its invention in the 17th century, it stood on a shaky ...

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.

In multivariable calculus, functions can depend on multiple variables. Evaluating these functions involves substituting values for all the variables. For example, if f (x, y) = x² + ...