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Let $f \colon U\subset \mathbb{R^n}\to\mathbb{R}^n$ be an injective Darboux map. Does this imply that $f$ is an open map?

If $f$ is continuous then the result follows from "Invariance of domain".

My Question: Does "Invariance of domain" hold true for injective Darboux function (instead of continuous injection)?

ViktorStein
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Sourav Ghosh
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  • A similar question was asked in the past (maybe for bijections) and the expectation is that such maps are continuous. – Moishe Kohan Feb 12 '23 at 19:44
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    https://mathoverflow.net/questions/235893/does-there-exist-a-bijection-of-mathbbrn-to-itself-such-that-the-forward-m – Moishe Kohan Feb 12 '23 at 23:43

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