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): Uncle! (1st condition satisfied)
by Steve Herman)
Your example for the first part looks good. Applying something like it to satisfy the second part appears to be more difficult.
Addition: F(0) needs to be defined, say F(0) = 1.
Edited on August 18, 2006, 1:21 pm
Posted by Bractals
on 2006-08-18 11:25:20