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.

JLo, I didn't see where you mentioned "monotonic" at all in your last hint.

Anyway, I assume you mean your revised definition of monotonic is:

x<y implies that f(x) <= f(y)

...instead of f(x) < f(y).  Is that correct?