Weierstrass function

In mathematics, the Weierstrass function, named after Karl Weierstrass, is a special kind of function. It is continuous everywhere but differentiable nowhere. This means the function has no breaks, but 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. It was created to show that not every continuous function is smooth. This changed how mathematicians thought about smoothness.

Today, the Weierstrass function is important in areas such as models of Brownian motion. These models need paths that are infinitely jagged, and are now known as fractal curves.

Construction

The Weierstrass function is a special kind of mathematical function. It is continuous everywhere, but it 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. This means it adds up an endless number of smaller parts to create the whole. Even though it cannot be differentiated anywhere, the function stays continuous. This means it has no breaks or jumps. The Weierstrass function is also one of the earliest examples of a fractal. A fractal is a shape that shows detail at every level when you zoom in.

Riemann function

The Weierstrass function is linked to an older idea called the Riemann function. This function was once thought to have no clear slope anywhere. But Riemann never wrote proof of this, and Weierstrass also didn’t find any proof in Riemann’s work.

Later, other mathematicians looked more closely at the Riemann function. In 1916, G. H. Hardy showed that the function does not have a derivative at some points related to pi (π). In 1969, Joseph Gerver found that the function does have a derivative at other special points also related to pi. Since then, more has been learned, showing that the Riemann function only has a derivative at just a few points.

Hölder continuity

The Weierstrass function can be written in a special way. This helps us see how slowly or quickly the function changes. Mathematicians call this kind of change "Hölder continuous."

For this function, there is a number K. It helps us measure how much the function changes between any two points. This makes the Weierstrass function interesting. It is smooth in one way, but not smooth in another way. This shows 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. 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