Let $f \colon \mathbb{R} \to \mathbb{R} $ be a continuous function so that $ f(x) \in \mathbb I = \mathbb R\setminus \mathbb Q,\ \forall x \in \mathbb{R}$. Prove that $f$ is constant.

I tried assuming it's not a constant but I can't get to a contradiction with continuity.

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  • 2
    What's $\mathbb{I}$? – Dante Grevino Nov 19 at 12:26
  • @DanteGrevino it's set of irrational numbers – user560461 Nov 19 at 12:27
  • 5
    Hint: assume it is not and then use the Intermediate Value Theorem to show that $f$ must take on a rational value. – John Douma Nov 19 at 12:29
  • 1
    Basically, rationals are everywhere. You cannot move an inch, a millimeter or an angstrom without stepping on a rational, or, for that matter, an infinity of rationals. – Eric Duminil Nov 19 at 15:50

Assuming we know 1. The Intermediate Value Theorem and 2. that between any two irrationals there is a rational, then the proof by contradiction should work fine.

If $f(a)<f(b)$ then there has to be a rational number $r$ with $f(a)<r<f(b).$ By IVT, there is a $c$ such that $f(c)=r$, contradiction.

The image $f(\mathbb{R})$ is path-connected and the only non-empty path-connected subspaces of $\mathbb{I}$ are the points. So $f$ is constant.

The continuous image of a connected set is connected.

The only connected sets in $\mathbb R\setminus \mathbb Q$ are single points.

Done.

Suppose that $f$ is not constant. Then there are $i_1, i_2 \in \mathbb I$ such that $i_1 \ne i_2$ and $i_1,i_2 \in f(\mathbb R)$. WLOG we can assume that $i_1 <i_2$. Now pick some rational number in $(i_1, i_2)$. The intermediate value theorem gives $r \in f(\mathbb R)$, a contradiction.

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