Weierstrass function

In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is a special kind of function that is continuous everywhere but differentiable nowhere. This means that while the function has no breaks or jumps, it is not smooth at any point — it has no defined slope anywhere. It is also an example of a fractal curve, which looks the same no matter how much you zoom in.

The Weierstrass function was first published in 1872 and was created to challenge the idea that every continuous function is smooth except at a few isolated points. Weierstrass showed that continuity does not always mean that a function can be differentiated almost everywhere, which changed how mathematicians thought about smoothness. At the time, these kinds of functions were hard to accept because they were difficult to visualize.

It wasn’t until computers became common that people could see what these functions looked like. Today, the Weierstrass function is important in areas such as models of Brownian motion, where infinitely jagged paths are needed. These paths are now known as fractal curves.

Construction

The Weierstrass function is a special kind of mathematical function that is continuous everywhere but cannot be differentiated at any point. It was first presented by Karl Weierstrass in 1872 to the Königliche Akademie der Wissenschaften.

This function is built using an infinite series, which means it adds up an endless number of smaller parts to create the whole. Despite not having derivatives anywhere, the function remains continuous, meaning it has no breaks or jumps. The Weierstrass function is also considered one of the earliest examples of a fractal, a shape that shows detail at every level of magnification.

Riemann function

The Weierstrass function is related to an earlier idea called the Riemann function. This function was thought to be differentiable nowhere, meaning it didn’t have a clear slope at any point. However, Riemann never published proof of this, and Weierstrass also didn’t find any evidence in Riemann’s work.

Later mathematicians studied the Riemann function more closely. In 1916, G. H. Hardy showed that the function does not have a derivative at certain points involving pi (π). In 1969, Joseph Gerver discovered that the function does have a derivative at other specific points also involving pi. More discoveries followed, showing that the Riemann function is differentiable only at very few points.

Hölder continuity

The Weierstrass function can be written in a special way that helps us understand how smooth or rough it is. This special way shows that the function changes very slowly, which mathematicians call "Hölder continuous."

For this function, there is a number K that helps us measure how much the function changes between any two points. This makes the Weierstrass function interesting because, while it is smooth in one way, it is not smooth in another way, showing us new ideas about how functions can behave.

Main article: Hölder continuous

Lipschitz continuous

Density of nowhere-differentiable functions

The Weierstrass function is not alone in its special properties. In fact, most continuous functions are like it — they are nowhere-differentiable. This means that if you look at all the smooth, continuous paths you can draw between 0 and 1, the ones that are not smooth anywhere are actually the most common.

Think of it this way: if you pick a continuous function at random, it is very likely that you will end up with one that, like the Weierstrass function, cannot be sloped or tilted at any point. This shows just how unusual smooth, differentiable functions really are!

Main article: Nowhere-differentiable function