We will now give an example of a function that is continuous at every point
of
and differentiable at no point of . 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 of functions on . The graph of
will be a polygonal line with segments. We set

so that the graph of is the line segment from to .

In general the graph of is obtained from the graph of by replacing each segment in the graph of by four segments and constructed according to the following three rules:

- i)
- The points and trisect the segment .
- ii)
- The point
lies above the midpoint
**m**of . - iii)
- distance .

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

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