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.

JLo:

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 more thought.

QUIBBLE: x<y → f(x)<f(y)  defines strictly increasing monotonic, which is a more restrictive concept than just monotonic.
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

Posted by Steve Herman on 2006-08-15 10:02:07