Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?

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This event attributed to Claude Fable 5.1

This event attributed to Peter L.

This event attributed to GPT-6 (Codex, Ultra effort)

VibeMathed record: Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not? evidence for this event

Answered in the negative for every $n\ge 2$. The preprint constructs a bijection $F:\mathbb R^n\to\mathbb R^n$ that maps every connected set to a connected set, is continuous exactly off the closed ray $[0,\infty)\times\{0\}^{n-1}$, and pulls the straight segment $\{(1,0,\dots,0)\}\times[0,1]$ back to the middle-thirds Cantor set on that ray, so $F^{-1}$ is not connectedness-preserving. The construction extends a thin solid tube by finger moves so its cross-sections recur near every point of the complementary compactum, collapses the ray onto the tube's ideal end, and certifies arbitrary connected sets by a separation argument; $F$ and $F^{-1}$ can be taken Borel. The same author's companion note on Darboux injections from closed manifolds (Banakh-Banakh Problems 1.7 and 1.8) is a separate result and belongs in its own entry. parent of this event

Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not? parent of this event

Banakh and Banakh, The continuity of Darboux injections between manifolds (2020), which calls the problem still open evidence for this event

Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not? evidence for this event

Wong's question on MathOverflow (2016), with the five partial answers evidence for this event

Proof of Banakh-Banakh's Problems 1.7 and 1.8 evidence for this event