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.
One might also define a function with an upwards jump of exp(-n) at each rational m/n (m, n relatively prime) and an additional jump at 0. It has the beautiful property of the w function from previous posts, namely that the larger the jumps, the rarer they are. It is continuous at the irrationals because the size of the jumps decreases in their vicinity (as n grows), and is finite because the sum of n * exp(n) converges.
Posted by vswitchs
on 2006-08-25 18:06:37