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 Uncle!
by Steve Herman)
Correct, there must be a jump discontinuity at every rational number. Just make sure the jumps add up to something finite. Fortunately the rational numbers are countable.
To get started, try to find a function f:[0,1]->R that is discontinuous in every number with finite binary representation. It is only a small step from there to the original problem.
Posted by JLo
on 2006-08-16 17:34:05