### CND Functions: Curves that are Continuous but Nowhere Differentiable

Approximation ${W_{12}(x)}$ to the Weierstrass CND function.

A function ${f(x)}$ that is differentiable at a point ${x}$ is continuous there, and if differentiable on an interval ${[a, b]}$, is continuous on that interval. However, the converse is not necessarily true: the continuity of a function at a point or on an interval does not guarantee that it is differentiable at the point or on the interval.

Throughout the eighteenth and early nineteenth centuries, mathematicians believed that continuity implied differentiability. The French physicist and mathematician André-Marie Ampère produced a result that, for a time, was known as Ampère’s Theorem: Every continuous function is differentiable except for a set of isolated points. To great surprise, this was shown to be false when Karl Weierstrass first described, at an address in 1872 to the Berlin Academy, a function that was continuous everywhere but differentiable nowhere.

We will denote such functions as CND functions (for continuous nowhere differentiable functions). The result of Weierstrass was not warmly greeted: most mathematicians considered it artificial and “pathological”. The biography of Weierstrass on MathTutor (O’Connor and Robertson) describes the reaction thus: “Analysts who depended heavily upon intuition for their discoveries were rather dismayed at this counter-intuitive function.” But the discovery of CND curves like that of Weierstrass forced mathematicians to reconsider their definitions, and led to significant developments in analysis. The function of Weierstrass was the first published example of what is now known as a fractal curve.

In his Master’s thesis at the Luleå University of Technology, Johan Thim (2003) included a long list of CND functions, with the names of more than thirty mathematicians, among whom were Bolzano, Riemann, Weierstrass, Hankel, Darboux, Sierpinski, Hardy, van der Waerden and Banach. The earliest of these was Bernhard Bolzano who, around 1830, constructed the first example of a CND function. However, it was not published for another century and, like much of his work, failed to gain the recognition that it deserved.

Weierstrass’s Function

Approximations for ${N=1,2,3,4}$ to the Weierstrass CND function.

The function devised by Weierstrass is defined as the limit of a sequence:

$\displaystyle W(x) = \sum_{k=0}^\infty a^k \cos b^k\pi x \,,$

where ${0, ${b>1}$ and ${ab\ge 1}$. We define the finite approximations to ${W(x)}$ by

$\displaystyle W_N(x) = \sum_{k=0}^N a^k \cos b^k\pi x \,.$

In the Figure above, approximations for ${N=1,2,3,4}$ are shown for parameter values ${a=\frac{1}{2}}$ and ${b=3}$. We see that the course of ${W_N(x)}$ becomes more erratic as ${N}$ increases. The graph of ${W_{12}(x)}$ is shown at the head of this post.  It gives a clear impression of the spiky nature of the limiting curve ${W(x)}$.

A Feast of Fractals

Approximations to the Koch snowflake curve.

In 1890, Giuseppe Peano defined a CND function that had a remarkable property: it was a space-filling curve which mapped the unit interval ${[0,1]}$ onto the unit square ${[0,1]\times[0,1]}$. Peano’s function was continuous but nowhere differentiable on the unit interval. It gave rise to the discovery of many other similar curves, an early one being by David Hilbert, who showed the everywhere continuity and nowhere differentiability of his curve [see That’s Maths post].

Another early example of a fractal curve was the Koch snowflake, which appeared in a 1904 paper by Swedish mathematician Helge von Koch, entitled On a continuous curve without tangents, constructible from elementary geometry. This curve is constructed by repeatedly trisecting the sides of an equilateral triangle and erecting a smaller triangle on the central segment. The first four stages are shown in the Figure (above left).

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Sources

${\bullet}$ Thim, Johan, 2003: Continuous Nowhere Differentiable Functions. Master’s Thesis, Dept of Maths., Luleå Univ. Technology, 2003:320 CIV. PDF

${\bullet}$ O’Connor, J.J. and E.F. Robertson. Karl Weierstrass — Biography. MathTutor,

${\bullet}$ That’s Maths: Space-Filling Curves, Part I: “I see it, but I don’t believe it”.