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: Nice solution!!! Now, you Want to try this by Ken Haley)
Ken,
actually I was asking you to construct a function that has prescribed jumps (i.e. left limit minus right limit) p^2 at the rationals. The solution I had in mind can achieve this easily. Now posted as requested.
That makes me wonder, did you try to figure out exactly which jumps your function has (The one with domain ]0,1[)? If you order all the jumps from large to small, you must end up with a converging series, I am just curious what the series looks like.

Posted by JLo
on 20060825 11:31:07 