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A converse of sorts to the intermediate value theorem, with an additional property
Definition of Darboux function: Let $S\subset\mathbb{R}$ be given. We say that $f:S\rightarrow\mathbb{R}$ is a Darboux function if it possesses the following property: for each $a,b\in S$, such that $a
My problem is this: Suppose that $f:\mathbb{R}\rightarrow\mathbb{R}$ is a Darboux function. Suppose also that $f^{-1}(\{q\})$ is closed for each $q\in\mathbb{Q}$. I have to prove that $f$ is continuous on $\mathbb{R}$.
Could you help me with this problem, please?