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

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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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asked Feb 12, 2023 at 18:07

Sourav Ghosh

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$\begingroup$ A similar question was asked in the past (maybe for bijections) and the expectation is that such maps are continuous. $\endgroup$
– Moishe Kohan
Feb 12, 2023 at 19:44
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$\begingroup$ mathoverflow.net/questions/235893/… $\endgroup$
– Moishe Kohan
Feb 12, 2023 at 23:43

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