The famous everywhere continuous, nowhere differentiable functions: van der Waerden’s and others

Abstract

One of the most weird functions is that which is defined for all real x, is continuous at each x but is differentiable at no x. Geometrically, it would appear to be some kind of limit of the saw tooth function,

provided the limit does not degenerate into a straight line. The first example of a continuous nowhere differentiable function was given by Weirstrass (1815–1897), namely $f(x) = \sum\nolimits_{n = 0}^\infty {{b^n}\cos ({a^n}\pi x)}$, where b is an odd integer and a is such that 0 < a < 1 and $ab > 1 + \frac{3} {2}\pi$. Now there are many elegant constructions of such functions (sometimes called the Weierstrass functions or the blancmange functions, see [109]); some very geometric, others very analytical (i.e., pictures very difficult to visualize), yet others a compromise. We begin with a delightfully simple example given by van der Waerden. The function is simply

$$\Phi \left( x \right) = \sum\limits_{k = 0}^\infty {{\phi _k}} \left( x \right) = \sum\limits_{k = 0}^\infty {\frac{1}{{{2^k}}}} {\phi _0}\left( {{2^k}x} \right),$$

, where ϕ₀(x) is the distance of x to the nearest integer.