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

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.

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.