# [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
```