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What is a creepy or weird fact that would scare even the bravest person?
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Your explanation is wrong. There is no reason to believe that "c" has no mapping.
Yeah, OP seems to be assuming a continuous mapping. It still works if you don't, but the standard way to prove it is the more abstract "diagonal argument".
Give me an example of a mapping system for the numbers between 1 and 2 where if you take the average of any 2 sequentially mapped numbers, the number in-between is also mapped.