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