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# 12.2 A Nowhere Differentiable Continuous Function.

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 .
The graphs of and are shown above . It can be shown that for each the sequence converges. Define on by

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.

Next: 12.3 Maxima and Minima Up: 12. Extreme Values of Previous: 12.1 Continuity   Index
Ray Mayer 2007-09-07