In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Weierstrass function is an example of a real-valued
function that is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
everywhere but
differentiable nowhere. It is an example of a
fractal curve. It is named after its discoverer
Karl Weierstrass.
The Weierstrass function has historically served the role of a
pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of
smoothness. These types of functions were denounced by contemporaries:
Henri Poincaré famously described them as "monsters" and called Weierstrass' work "an outrage against common sense", while
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
...
wrote that they were a "lamentable scourge". The functions were impossible to visualize until the arrival of computers in the next century, and the results did not gain wide acceptance until practical applications such as models of
Brownian motion necessitated infinitely jagged functions (nowadays known as fractal curves).
Construction
In Weierstrass's original paper, the function was defined as a
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
:
:
where