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.
AWE: Yes, this sounds like an odd function. I am
aware of a function that is continuous at every rational number,
discontinuous elsewhere, but it is not monotonic. This requires
QUIBBLE: x<y → f(x)<f(y) defines
strictly increasing monotonic, which is a more restrictive concept than
f monotonic increasing only requires that x<=y → f(x)<=f(y).
For instance, f(x) = 0 is monotonic, but not strictly monotonic.
Edited on August 20, 2006, 6:59 pm