Find a function f:R->R (R the set of real numbers), such that
1. f has a discontinuity in every rational number, but is continous everywhere else, and
2. f is monotonic: x<y → f(x)<f(y)
Note: Textbooks frequently present examples of functions that meet only the first condition; requiring monotonicity makes for a slightly more challenging problem.
(In reply to re(2): Bigger hint (and proposed problem redefinition)
by Ken Haley)
You've done it! Congratulations all around.
A formulation I like a little better (using Ken's) notation is:
W(x) = w(x) + sign(x)
where sign(x) = 1 if x > 0
-1 if x < 0
0 if x = 0
This has the pleasing quality that W(x) = -W(-x).
But I'm just gilding the lily, now.
Great problem and great hints, JLO!
Great contribution, Ken!
I'm rating this a 5 also!
And I withdraw my earlier suggestion about redefining the problem to only consider positive real numbers.