[FOM] Category and Measure
Timothy Y. Chow
tchow at alum.mit.edu
Tue Sep 18 17:49:17 EDT 2007
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
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