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It is known that there is a bijection between rational numbers and finite continued fractions, so every rational number is uniquelly identified by a finite continued fractions and vice versa. It is also known that for any irrational number, we can find an infinite continued fractions, but I don't have information is it a unique. If so, than we can conclude that there is a bijection between real numbers and continued fractions.

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    $\begingroup$ Wikipedia says it's unique for irrationals. But you don't really need that. There can't be more continued fractions than $\aleph_0^{\aleph_0}=2^{\aleph_0}=|\mathbb R|$ $\endgroup$ – user2345215 Mar 9 '14 at 12:53
  • $\begingroup$ For rationals, it isn't a bijection: each rational has two expansions. $\endgroup$ – GEdgar Mar 9 '14 at 14:06
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Only if the numerator of each such nested fraction is $1$. But if you're inquiring about generalized continued fractions, then the answer is obviously no.

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