Consider a variant on Example 1.7 of Chapter 1.  C_1 is defined
by removing the middle 1/3 of the unit interval.  C_2 is defined
by removing the middle 1/6 of each remaining interval (e.g., removing
two intervals of absolute size 1/18).  C_3 removes the middle 1/12
from each of the four remaining intervals, and so forth.  C is 
the intersection of these.  The measure of C is 

	b := (2/3) x (5/6) x (11/12) ... = 0.46798587... > 0 .

The function f := (1/b) 1_C (the indicator function of C normalized
by 1/b) has the property that F = integral of f (in the Lebesgue sense)
and f = F' in the sense of distributions, but F is nowhere differentiable
in the ordinary sense.  If you don't know the meaning of differentiability
in the sense of distributions, don't worry, it is not relevant here.
[OK, if you're really curious, it means that for all smooth functions
h vanishing outside a compact interval, the integral of F h' is equal
to the integral of f h.]