In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example …

The function was published by Weierstrass but, according to lectures and writings by Kronecker and Weierstrass, Riemann seems to have claimed already in 1861 that the function is not differentiable …

The function appearing in the above theorem is called the Weierstrass function. Before we prove the theorem, we require the following lemma: Lemma (The Weierstrass M-test). Let (E; d) be a metric …

May 18, 2025 · Discover the origins, properties, and significance of the Weierstrass function in real analysis. Explore its fractal nature and applications.

Proof. First we show the function f is continuous. Indeed, since jbk cos akxj bk and b 2 (0; 1), we know the series de ning f converges absolutely and uniformly on R by Weierstrass M-test. Moreover, since …

Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere". Here is an example of one: It is not hard to show that this series converges for all x. In fact, it is absolutely …

But the only holomorphic functions on a compact Riemann surface are the constant functions; since the constant term of the Laurant series of f (z) is 0, we conclude f (z) = 0 identically.

does not exist as a real number or 1. This rules out the possibility of the Weierstrass function having a vertical tangent line, or an “infinite derivative” anywhere.

In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass.

Until Weierstrass published his shocking paper in 1872, most of the mathematical world (including luminaries like Gauss) believed that a continuous function could only fail to be differentiable at some …