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 2006-08-18 18:36:17