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.
Study the following function g:[0,1]>R which is also pretty weird, although it does not exactly solve the problem:
Write x in [0,1] in binary notation. You obtain f(x) by interpreting this string as a number being written in ternary notation, i.e. with digits 0, 1 and 2. Instead of ternary notation you may use decimal notation, although then the function does not look that nice.

Posted by JLo
on 20060818 18:36:17 