Find a continuous, strictly monotonic function f:R->R (R the set of real numbers) which is non-differentiable on a very dense set.

For this problem, we'll call a set of real numbers very dense if it intersects every interval [a,b] in an infinite, uncountable number of elements.

(In reply to re(3): Not dense enough ... by vswitchs)

I'm afraid the midpoint displacement construction (see previous posts) doesn't work after all.  As I wrote in my last post, to get a bound for the derivative of the limit function to allow to make it monotonous by adding a linear function, one would have to make the successively added functions smaller and smaller in a way which allows the sum of their slopes to converge.  But then, the limit function will become differentiable at the irrationals: Every irrational is contained in nested intervals of the form [a*2^-n, (a+1)*2^-n].  The limit of difference quotients between the endpoints of these intervals converges for n->infinity.  Admittedly this alone does not prove differentiability.  If there exists a different series of difference quotients which converges to a different value, or not at all, the function would still be non-differentiable at the irrationals.  But I haven't been able to come up with one.  So I think said function would be differentiable at the irrationals, and no solution to our problem.

Note to JLo: Can you confirm that there is a solution?  I'm thinking of the problem of a lexicographical order for the real functions, which turned out to have no known solution.  Sort of unsatisfactory, that...

Posted by vswitchs on 2006-10-15 06:59:28