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Let $f_n: \mathbb{R} \to \mathbb{R}$ for $n \in \mathbb{N}$ be continuous function. Suppose that for any $x \in \mathbb{R}$ there is some $n$ such that $f_n(x) \in \mathbb{Q}$. Prove that for every $c < d$ in $\mathbb{R}$, one can find some numbers $a < b$ in the interval $(c,d)$ and $n \in \mathbb{N}$ such that $f_n$ is constant on $(a,b)$.

My intuition for this question is that I would want to use something like the intermediate value theorem (a related question here does that)? But I'm not sure how everything goes together.

MathFrak96
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1 Answers1

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Fix $[c,d]$ enumerate rationals, say by $q_m$. Now, for any $n$ and any $m$ the set $A_{n.m} = f_{n}^{-1}(q_m)\cap [c,d]$ is closed (because functions are continuous). From Baire's category theorem and assumptions on the functions $[c,d]= \cup{A_{n,m}}$ so one of $A_{n,m}$ has to have non empty interior, hence it contains sub-interval $[a,b]$.

Salcio
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