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(2): Not dense enough ... by Steve Herman)

I think the function at
http://www.math.tamu.edu/~tom.vogel/gallery/node7.html
has an unbounded derivative. For instance, at 0, you add up an infinite number of linear pieces with slope 1, which gives an infinite derivative for the limit function at 0.  So one can't make it monotonic by adding a linear function c*x.

But one could fix this by, instead of forming the sum of g_k(x)  (using the nomenclature at the web page), summing up 2^(-k) * g_k(x).  The maximum derivative would then be the sum over k of 2^(-k), which is finite.  I think this would make the limit function differentiable at some points.  But if the remaining points were uncountable, it would be a solution. (Or rather, it plus a linear function).

Posted by vswitchs on 2006-10-13 16:16:25