The Weierstrass function

Until the middle of the nineteenth century almost every mathematician took for granted that a continuous function had to be differentiable “almost everywhere”: at most it could have a few isolated corners, like $f(x)=|x|$ at the origin. Geometric intuition — a graph drawn without lifting the pen seems to have a tangent at every point — appeared to confirm this.

In 1872 Karl Weierstrass presented to the Berlin Academy a startling counterexample: a function continuous at every point of the real line but differentiable at no point. His construction is a series of functions: $W(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x),$ where $0 < a < 1$ and $b$ is an odd integer chosen so that $ab > 1 + \tfrac{3}{2}\pi$. The condition $a<1$ makes the series totally convergent, so $W$ is continuous; but the factor $b^n$, growing without bound, introduces ever finer oscillations at every scale, so that the difference quotient never converges to a finite value. The graph is self-similar: magnifying any portion reveals the same jagged irregularity, foreshadowing fractals.

The result had the effect of an earthquake. It showed that visual intuition was not a reliable guide and that analysis had to rest solely on rigorous definitions, such as those based on $\varepsilon$ and $\delta$. It was later discovered that already around 1830 Bernhard Bolzano had constructed a similar example, which nevertheless remained unpublished for decades (Boyer, Katz).

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Topics: Continuita Concepts: Continuita People: Karl Weierstrass · Bernard Bolzano