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Weird function challenge (Posted on 2006-08-15) Difficulty: 4 of 5
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.

See The Solution Submitted by JLo    
Rating: 4.3000 (10 votes)

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Hints/Tips Bigger hint | Comment 7 of 33 |
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
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