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Can anyone give me an example of a function $f: \Bbb R \rightarrow \Bbb R$ such that for any open subset $V$ in $\Bbb R$, $f(V)$ is open but $f$ is not continuous at any point.

This was a side comment my professor made and I haven't been able to quite wrap my head around this yet.

Iff
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    you can see this http://math.stackexchange.com/questions/75589/open-maps-which-are-not-continuous and compose with an homeomorphism between $\Bbb R$ and $(0,1)$ – Bérénice Mar 30 '16 at 21:19

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Try the Conway base 13 function. The image of any open set is the entire real line, but...

Henricus V.
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