The mathematician Karl Weierstrass was born #OTD, Halloween, in 1815. The fools at the academy all said he was mad, but in 1872 he announced that he had succeeded in creating a ~monster~.

Image: Smithsonian Institution Libraries

Henri Poincare, who was first to understand the terrible thing Weierstrass had done, called it "an outrage against common sense.” Somehow, Weierstrass had stitched together and brought to life a pathological function made of nothing but pointy teeth and sharp corners.

"Weierstrass's Monster," as Poincare called it, is a function that is everywhere continuous but nowhere differentiable.

Here are three plots of the first 1000 terms in the sum ( over [-2,2], [-0.2,0.2], and [-0.02,0.02] ) using the values a=1/3 and b=21 from Weierstrass's original proof.

Do you see? No matter how far you zoom in it's all sharp corners. Any finite stretch along the x-axis opens up into a terrible mouth of uncountable teeth, every pointed sliver becomes a thousand little razors.

Mathematicians were aghast.

Using only accepted notions of infinite sums, limits, derivatives, and continuity, Weierstrass had assembled something unholy.

Hermite said of Weierstrass's construction: "I turn with terror and horror from this lamentable scourge of functions with no derivatives."

Mathematicians recoiled, but they were forced to admit that Weierstrass's conclusions were sound. Their intuition, which assumed a connection between continuity and some degree of smoothness, now had to accommodate the existence of monsters.

Weierstrass was very pleased with himself. The fools at the Academy had called him mad, but after the 1872 presentation many mathematicians discovered that they, too, could bring monsters to life. Similar constructions began to appear everywhere.

In fact, Banach later proved that differentiable functions are a "meager set" of all continuous functions.

Differentiability is a fairy tale we tell our students to protect them from the horrible fact that most continuous functions are monsters.

The monsters are everywhere.

" ... but now that I had finished, the beauty of the dream vanished, and breathless horror and disgust filled my heart."
– Victor Frankenstein

Happy Halloween!

Actually, “Weierstrass” was the name of the Doctor of Mathematics who found the function, and I, the person who made a long thread out of it, am the Monster.