Does there exist a continuous onto function from $\mathbb{R}-\mathbb{Q}$ to $\mathbb{Q}$? (where domain is all irrational numbers)
I found many answers for contradicting the fact that there doesnt exist a continuous function which maps rationals to irrationals and vice versa.
But proving that thing was easier since our domain of definition of function was a connected set, we could use that connectedness or we could use the fact that rationals are countable and irrationals are uncountable.
But in this case those properties are not useful. I somehow think that baire category theorem might be useful but I am not good at using it.