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: Bigger hint (and proposed problem redefinition)
by Steve Herman)
Wow. Nice job, Steve...and great problem, JLo.
Can't we just fix the zero problem as follows? Let's call JLo's example function w (for "weird"). Define the solution function, W (upper-case) as
W(x) = w(x) for x>0
= -1 for x=0
= w(x) - 2 for x<0
This gives us the needed discontinuity at 0 and maintains the characteristics of w(x) everywhere else, and the whole function is monotonic if w(x) is.
In any case...I rate this puzzle a 5!