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.
Sorry to keep everybody waiting. My idea, which doesn't quite
work, involves starting with f(x) = x, and then operating on the
fractional part of each one unit range as follows:
f'(x) = the integer part of x + g(fractional part of x)
where g(x) =
if x <= .5 x/2
if x > .5 (3x+1)/2
In effect, I have pulled the function down a little at the middle of
each unit range. This function is continuous, strictly monotonic,
and non-differentiable wherever x = n/2, n being an integer.
This process can be continued indefinitely, by "pulling down" the
midpoint of each line segment. The limiting function can be
evaluated for any number by converting the number to base 2 and
performing an iterative operation that I won't describe here. The
limiting function is continuous and strictly monotonic. It is
non-differentiable for any number of the form n/2^k, but I
unfortunately think that it is differentiable for any irrational
number. And the numbers of the form n/2^k are countable on any
interval ...
Edited on October 8, 2006, 10:01 am
Edited on October 8, 2006, 10:03 am
Not dense enough ...
