In the wikipedia article about 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 that it is actually homeomorphic to the irrational numbers with their usual subspace topology iherited from the real line.

So my questions:

Can we describe explicitely a surjective continuous map from the irrationals to $\mathbb{R}$?

Same for a continuous bijection from a closed subset of the irrationals to $\mathbb{R}$.