Continuous Nowhere-differentiable Functions

Over the past few weeks, we have talked about the continuity and differentiability of a function and we want to intuitively related these two concept with each other because they all characterize some important properties of a function. In Chapter 5 of Understanding Analysis by Abbott, there is a theorem states that if $f:A\to\mathbb{R}$ is differentiable at a point, then it must be continuous at that point as well. We also find that this property gets weakened from the other direction, where continuity does not necessarily imply differentiability. One popular example would be the absolute value function such that $f(x)=|x|$ , where f is continuous but not differentiable at its cusp $x=0$ .

Another example could be the cube root function $f(x)=x^{1/3}$ at $x=0$ , where f has a vertical tangent line that is not defined.

You may want to ask (or guess from the topic up front): does there exist a function that is continuous but not differentiable at any point? The answer is absolutely yes. And in fact, among all real-valued continuous functions, the subset of all continuous but nowhere differentiable functions is quite “large”.

A Glance of History

The first example of a continuous nowhere differentiable function on an interval is due to Czech mathematician Bernard Bolzano originally around 1830, but not published until 1922. Unlike many other constructions of nowhere differentiable functions, Bolzano’s function is based on a geometrical construction instead of a series approach. The Bolzano function, $B$ , is constructed as the limit of a sequence { $B_k$ } of continuous functions.

The first published nowhere differentiable continuous function – Weierstrass Function – is presented by Karl Weierstrass on July 18, 1872. The function is given of the form (modified by Hardy in 1916):

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

for $0<a<1, ab\ge1~and~b>1$ . The function is an example of a Fourier Series, a very important and fun type of series. Here, for the sake of better understanding, we simplify some details of Weierstrass’s argument by replacing the cosine function with a piecewise linear function that holds a similar property with $cos(x)$ when it comes to proving it is continuous and nowhere differentiable.

A little “pf”

Let’s define the function $h:\mathbb{R}\to\mathbb{R}$ such that $h(x)=|x|$ where $x\in[-1,1]$ and $h(x+2)=h(x)$ where $x\in\mathbb{R}$ . So we can see from the figure below that $h$ is periodic of period 2.

Now we may define $f(x)=\sum_{n=0}^\infty (\frac{3}{4})^n h(4^n x)$

This might look familiar to you because it is of a similar form to the construction of Cauchy Condensation Test, but not quite. Actually, we may apply a Weierstrass M-test to show that the infinite series is continuous on $\mathbb{R}$ give that $|h(x)\le1|$ and $\latex M_n=(\frac{3}{4})^n$.

A little more difficult task is to show that $f(x)$ is non-differentiable on all of $\mathbb{R}$ . Ultimately, we want to show that the sequence ( $x_m$ ) converges to 0 such that $\frac{f(x_m)-f(x)}{x_m-x}$ diverges as $m\to\infty$ , which implies that $f'(x)$ does not exist. Applying some algebraic tricks here we will be able to get the result as desired.

More examples and interesting applications

Takagi presented his example in 1903 as an example of a “simpler” continuous nowhere differentiable function than Weierstrass. The definition of Takagi’s function is expressed as the infinite series:

$T(x) = \sum_{k=0}^\infty \frac{1}{2^k} dist(2^kx,\mathbb{Z}) = \sum_{k=1}^\infty \frac{1}{2^k}\inf_{m\in\mathbb{Z}} |2^k x-m|$

In 1904, Swedish mathematician Helge von Koch published an article about a curve of infinite length with tangent nowhere. Koch writes:

Even though the example of Weierstrass has corrected this misconception once and for all, it seems to me that his example is not satisfactory from the geometrical point of view since the function is defined by an analytic expression that hides the geometrical nature of the corresponding curve and so from this point of view one does not see why the curve has no tangent.

Koch’s “snowflake” curve (named after its shape) is constructed as follows: Take an equilateral triangle and split each line in three equal parts. Replace the middle segments by two sides of a new equilateral triangle that is constructed with the removed segment as its base. Repeat this procedure on each of the four new lines (for each of the original three sides). Repeat indefinitely.

The continuous nowhere-differentiable functions also have a broad application in other scientific fields. One interesting note is that the well-known Brownian motion, or pedesis, has path that is continuous and nowhere differentiable. It is insightful to keep in mind that although the path of a Brownian motion is everywhere continuous, its randomness makes it nowhere differentiable. Therefore we actually cannot draw any conclusion on the moving direction in terms of differentiability. (Check out a gif simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random directions at https://upload.wikimedia.org/wikipedia/commons/c/c2/Brownian_motion_large.gif.)

References

https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter06/section03.html

https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf

http://www.math.ubc.ca/~feldman/m321/nondiffble.pdf

https://en.wikipedia.org/wiki/Brownian_motion

Math 320: Real Analysis