Take the tour ×  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.   share|improve this question See Problems in real
and complex analysis By Bernard R. Gelbaum, pages 17 and 155. – 
Martin Sleziak Oct 19 '11 at 18:20 Comment (I don't have enough
points to post a comment, sorry): A delta-epsilon proof is not
too hard. –  gary Oct 19 '11 at 20:49 add comment  1 Answer  Let
$f$ be the Cantor function. Let $y\in[0,1]$. Let
$0.d_1d_2d_3\ldots$ be the binary expansion of $y$. Let $x$ be
the number whose ternary expansion is
$0.(2d_1)(2d_2)(2d_3)\ldots\ {}$. Then $f(x)=y$. For numbers
with non-unique binary expansions, one gets two ternary
expansion