In the Wikipedia article about [descriptive set theory](https://en.wikipedia.org/wiki/Descriptive_set_theory) I read that $\mathbb{R}$ (with its usual topology) is a Polish space, and that every Polish space 


1) can be obtained as a continuous image of the Baire space $\mathcal{N}$


2) can be obtained as the image of a continuous bijection defined on a closed subset of the Baire space. 


Then I learn from the Wikipedia article on [Baire space](https://en.wikipedia.org/wiki/Baire_space_(set_theory)) that it is actually homeomorphic to the irrational numbers with their usual subspace topology inherited from the real line.


So my questions:



> 
> Can we describe explicitly a surjective continuous map from the irrationals to $\mathbb{R}$?
> 
> 
> Same for a continuous bijection from a closed subset of the irrationals to $\mathbb{R}$.
> 
> 
> 



The reals as continuous image of the irrationals
For any irrational number $x$, let $f(x)$ be the real number arising from the integer part of $x$, together with every other digit of the rest of the expansion of $x$. 


This is surjective, since one may interleave the digits of any real $y$ with any nonrepeating pattern, and thereby find an irrational $x$ with $f(x)=y$. This is continuous, since if $x\_n\to x$, then $f(x\_n)\to f(x)$.


