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 Not dense enough ... by Steve Herman)

I think your solution works, and is in fact differentiable nowhere.  The difference quotient (f(x)-f(x2))/(x-x2) oscillates indefinitely due to the midpoint displacements introduced in successive stages of the construction process.

Similarly constructed non-monotonic functions are known to be differentiable nowhere, see for instance http://www.math.tamu.edu/~tom.vogel/gallery/node7.html .  All that remains to be taken care of is monotonicity.  I'm not entirely sure how that is done in your solution, Steve.  The slopes of the successively added linear pieces have to add up to something finite and bounded.  (Then one can add a linear function with a slope larger than that bound to make the result monotonic.)  Not sure whether this destroys the non-differentiability...  Have to think about it.

Posted by vswitchs on 2006-10-12 02:50:54