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)

Ken:

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.