Weierstrass function

In mathematics, the Weierstrass function is an example of a pathological real-valued function on the real line. The function has the property of being continuous everywhere but differentiable nowhere. It is named after its discoverer Karl Weierstrass.

Historically, the Weierstrass function is important because it was the first published example (1872) to challenge the notion that every continuous function was differentiable except on a set of isolated points.

In Weierstrass' original paper, the function was defined as the sum of a Fourier series:

where ${\displaystyle 0<a<1}$, ${\displaystyle b}$ is a positive odd integer, and

The minimum value of ${\displaystyle b}$ which satisfies these constraints is ${\displaystyle b=7}$. This construction, along with the proof that the function is nowhere differentiable, was first given by Weierstrass in a paper presented to the Königliche Akademie der Wissenschaften on 18 July 1872.

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