[FOM] Category and Measure

[Timothy Y. Chow]

On Tue, 18 Sep 2007, joeshipman at aol.com wrote:
> Can anyone point to an explicit example of an everywhere differentiable 
> function such that the meager set of points where the derivative fails 
> to be continuous is not measure 0 and therefore not Riemann integrable? 
> (Preferably a function whose derivative is bounded, since we want the 
> Riemann integrability to fail purely because of the measure-category 
> issue.)

See Chapter 8, number 35 of "Counterexamples in Analysis" by Gelbaum and 
Olmsted, Holden-Day, 1964.  They attribute an example similar to the one 
below to Volterra (1881).

Let g(x) = x^2 sin(1/x).  For any positive c, let x_c be the largest 
positive x <= c such that g'(x) = 0.  Define g_c(x) by

            / g(x)    if 0 < x <= x_c,
  g_c(x) = {
            \ g(x_c)  if x_c <= x <= c.

Let A be a Cantor set of positive measure in [0,1], and define f as 
follows: set f(x) = 0 if x is in A; otherwise, if x belongs to an interval 
(a,b) that was removed from [0,1] while forming A, then let c = (b-a)/2 
and set
            / g_c(x-a)   if a < x <= (a+b)/2,
    f(x) = {
            \ g_c(b-x)   if (a+b)/2 <= x < b,

Then one can show that f is everywhere differentiable on the unit interval 
and f' is bounded, but discontinuous at every point of A.

Tim