Ah, a favorite is a function that
is differentiable but its derivative
is not Riemann integrable!
Recall, a function is Riemann
integrable if and only if it
is continuous everywhere except
on a set of measure 0. So, the
derivative has to be discontinuous
on a set of positive measure.
Now, to construct one of those!
Here's another favorite: For positive
integer n and the set R of real numbers,
suppose C is a closed subset of R^n with
the usual topology. Then there exists
a function f: R^n --> R that is
0 on C, strictly positive otherwise,
and infinitely differentiable.
So, for C, take, say,
a sample path of Brownian motion,
the Mandelbrot set,
a Cantor set of positive measure,
etc. Can use that function to
settle an old question in
constraint qualifications for the
Kuhn-Tucker conditions in
optimization.
Or, any closed set can be the level
set of an infinitely differentiable
function.
Sure, really fun reading for such
things is:
Bernard R. Gelbaum and
John M. H. Olmsted,
Counterexamples in Analysis,
Holden-Day,
San Francisco,
1964.
No, it's doable at the level of
Rudin's Principles:
He shows that a function is
Riemann integrable if and only
if it is continuous everywhere
except on a set of measure 0.
For the definition of measure 0,
he gives that quickly, and don't
really need a course in measure
theory. Besides a course in
measure theory likes just to
f'get about Riemann integration,
and thankfully so.
Agree, it's do-able with baby Rudin. But as I recall it's a typical example used (early) in measure theory, not so much intro analysis. Hence "typical".
Comments
Ah, a favorite is a function that is differentiable but its derivative is not Riemann integrable!
Recall, a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0. So, the derivative has to be discontinuous on a set of positive measure. Now, to construct one of those!
Here's another favorite: For positive integer n and the set R of real numbers, suppose C is a closed subset of R^n with the usual topology. Then there exists a function f: R^n --> R that is 0 on C, strictly positive otherwise, and infinitely differentiable. So, for C, take, say, a sample path of Brownian motion, the Mandelbrot set, a Cantor set of positive measure, etc. Can use that function to settle an old question in constraint qualifications for the Kuhn-Tucker conditions in optimization.
Or, any closed set can be the level set of an infinitely differentiable function.
Sure, really fun reading for such things is:
Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964.
Fun book indeed!
Your first example you won't typically run into until an introductory measure theory course.
No, it's doable at the level of Rudin's Principles: He shows that a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0.
For the definition of measure 0, he gives that quickly, and don't really need a course in measure theory. Besides a course in measure theory likes just to f'get about Riemann integration, and thankfully so.
Agree, it's do-able with baby Rudin. But as I recall it's a typical example used (early) in measure theory, not so much intro analysis. Hence "typical".
Ymmv, of course.