If there is such a function then there is an injection from $\omega_1$ to $2^{\omega}$.
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(Set $\: X = \omega \:$, $\:$ send the finite ordinals to the corresponding singletons,
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then extend to $\omega_1$ with [transfinite recursion](http://planetmath.org/encyclopedia/TransfiniteRecursion.html).)

If there is such a function and the continuum hypothesis holds then $\: 2^{\aleph_0} = \aleph_1 \:$.

It is [consistent with ZF that the continuum hypothesis holds and](http://math.stackexchange.com/questions/43861/relationship-between-continuum-hypothesis-and-special-aleph-hypothesis-under-zf/43877#43877) $\: 2^{\aleph_0} \neq \aleph_1 \:$.

Therefore ZF does not prove the existence of such a function.