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How can I prove that the Cantor function is surjective and continuous?

The part, I think that the cantor function is monotonic and surjective, if I prove this, it is easy to prove that this implies continuity. The way to prove that is surjective, it's only via an algorithm, I don't know if this can be proved in a different way, more elegant. And the monotonicity I have no idea, I think that it's also via an algorithm.

Thanks

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    See Problems in real and complex analysis By Bernard R. Gelbaum, pages [17](http://books.google.com/books?id=v7_FiqY3FX0C&pg=PA17#v=onepage&q&f=false) and [155](http://books.google.com/books?id=v7_FiqY3FX0C&pg=PA155#v=onepage&q&f=false).2011-10-19
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    Comment (I don't have enough points to post a comment, sorry): A delta-epsilon proof is not too hard.2011-10-19

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