12.2 $^*$A Nowhere Differentiable Continuous Function.

We will now give an example of a function $f$ that is continuous at every point of $[0,1]$ and differentiable at no point of $[0,1]$. The first published example of such a function appeared in 1874 and was due to Karl Weierstrass(1815-1897) [29, page 976]. The example described below is due to Helga von Koch (1870-1924), and is a slightly modified version of Koch's snowflake. From the discussion in section 2.6, it is not really clear what we would mean by the perimeter of a snowflake, but it is pretty clear that whatever the perimeter might be, it is not the graph of a function. However, a slight modification of Koch's construction yields an everywhere continuous but nowhere differentiable function.

We will construct a sequence $\{f_n\}$ of functions on $[0,1]$. The graph of $f_n$ will be a polygonal line with $4^{n-1}$ segments. We set

$$f_1(x)=0\;\; \mbox{ for } \;\;0\leq x\leq 1$$

so that the graph of $f_1$ is the line segment from $(0,0)$ to $(0,1)$.

In general the graph of $f_{n+1}$ is obtained from the graph of $f_n$ by replacing each segment $[\mbox{{\bf a}}\mbox{{\bf e}}]$ in the graph of $f_n$ by four segments $[\mbox{{\bf a}}\mbox{{\bf b}}], [\mbox{{\bf b}} \mathbf{{\bf c}}], [\mathbf{{\bf c}}\mbox{{\bf d}}],$ and $[\mbox{{\bf d}}\mbox{{\bf e}}]$ constructed according to the following three rules:

i)

The points $\mbox{{\bf b}}$ and $\mbox{{\bf d}}$ trisect the segment $[\mbox{{\bf a}}\mbox{{\bf e}}]$.

ii)

The point $\mathbf{{\bf c}}$ lies above the midpoint m of $[\mbox{{\bf a}}\mbox{{\bf e}}]$.

iii)

distance $${({\bf m},\mathbf{{\bf c}})={{\sqrt 3}\over 2}\mbox{ distance}(\mbox{{\bf b}} ,\mbox{{\bf d}})}$$.

The graphs of $f_2,f_3,f_4$ and $f_{7}$ are shown above . It can be shown that for each $x\in [0,1]$ the sequence $\{f_n(x)\}$ converges. Define $f$ on $[0,1]$ by

$$f(x)=\lim \{f_n(x)\}\; \mbox{ for all }\; x\in [0,1].$$

It turns out that $f$ is continuous on $[0,1]$ and differentiable nowhere on $[0,1]$. A proof of this can be found in [31, page 168].

The function $f$ provides us with an example of a continuous function that is not piecewise monotonic over any interval.