The original version of This story appeared in Quanta Magazine.

Calculus is a powerful mathematical instrument. But hundreds of years after its invention in the 17th century, it stood on a shaky foundation. The core concepts were rooted in intuition and informal arguments, rather than precise, formal definitions.

According to Michael Barany, a historian of math and science at the University of Edinburgh, two schools of thought have emerged. French mathematicians were largely to move on. They were more concerned about the application of calculus to problems in physics – and used it to calculate the trajectories of planets, or to study the behavior of electrical streams. But by the 19th century, the German mathematicians began to break down things. They are planning to find counter -examples that would undermine long -terms, and eventually use the counter -examples to place Calculus on more stable and durable footing.

One of these mathematicians was Karl Refaced. Although he showed an early math plant, his father put him under pressure to study public finances and administration, with a view to the Prussian public service. Reflection, which is bored with his university course, spent most of his time drinking and fencing; In the late 1830s, after not obtaining his degree, he became a higher teacher and gave lessons in everything from math and physics to Penmanship and gymnastics.

Reflections first began his career as a professional mathematician before he was almost 40. But he would change the field by introducing a mathematical monster.

The pillars of calculus

In 1872, Rewrestrass published a function that threatened everything that mathematics thought they were understanding about Calculus. He received indifference, anger and fear, especially of the mathematical giants of the French Thinking School. Henri Poincaré condemned refusal’s function as “an indignation against common sense”. Charles Hermite calls it a ‘deplorable evil’.

Understanding why the result of refusals was so alarming helps to understand two of the most fundamental concepts in Calculus: continuity and distinction.

An ongoing function is exactly what it sounds like – a function that has no gaps or jumps. You can detect a path from any point on such a function to any other without lifting your pencil.

Calculus is largely about determining how quickly such continuous functions change. It works loosely speaking, by approaching a given feature with straight, non -performance lines.

Illustration: Mark Belan/Quanta Magazine