Is a path-connected bijection $f\colon \Bbb{R}^n \to \Bbb{R}^n$ continuous? While thinking about this question I was asking myself if a path-connected bijection $f\colon \Bbb{R}^n \to \Bbb{R}^n$ has to be continuous for $n>1$?
If we drop the requirement that $f$ is bijective, then it is not true as in the connected case. 
I was wondering if this question is maybe easier? I have no intuition if it is true or not. On one hand I think there are too many connected sets, there will be some ugly counterexample, on the other hand the reals are "nice". 
By a path-connected function I mean a function between topological spaces whose image of a path-connected set is path-connected. 
 A: Here's how you can get a counterexample for $n>2$ assuming the continuum hypothesis.  Note that if $A\subseteq\mathbb{R}^n$ is path-connected and $x,y\in A$, there is a path-connected closed subset $B\subseteq A$ containing $x$ and $y$ (namely, the image of a path between them).  So for $f$ to be path-connected, it suffices for $f(A)$ to be path-connected whenever $A$ is a closed path-connected subset of $\mathbb{R}^n$.
This is very useful, because there are only $\mathfrak{c}$ different closed subsets of $\mathbb{R}^n$.  First, partition $\mathbb{R}^n$ into $\mathfrak{c}$ sets $(S_\alpha)_{\alpha<\mathfrak{c}}$, each of which intersects every closed path-connected subset with more than one point at $\mathfrak{c}$ points (you can do this by an induction of length $\mathfrak{c}$, where at the $\alpha$th stage you add a point of the $\gamma$th closed path-connected set to $S_\beta$ for all $\beta,\gamma<\alpha$).  Also, enumerate all quadruples $(A,x,y,u)$ where $A$ is a path-connected closed subset of $\mathbb{R}^n$, $x,y\in A$, and $u\in\mathbb{R}^n$ with order-type $\mathfrak{c}$.
Now define $f$ by an induction of length $\mathfrak{c}$.  At the $\alpha$th stage of the induction, we want to define $f$ on $\{x,y\}\cup (A\cap S_\alpha)$ such that $f(A\cap S_\alpha)$ contains a smooth path from $f(x)$ to $f(y)$, where $(A,x,y,u)$ is the $\alpha$th quadruple in our enumeration.  To do this, we need the following lemma (here is where we use CH and the fact that $n>2$):

Lemma: Let $n>2$, let $a,b\in \mathbb{R}^n$ and let $g_1,g_2,\ldots:[0,1]\to\mathbb{R}^n$ be a countable collection of smooth paths and $c_1,c_2,\ldots\in\mathbb{R}^n$ be a countable collection of points distinct from $a$ and $b$.  Then there is a smooth path from $a$ to $b$ that does not intersect any of the $g_i$ except possibly at $a$ or $b$ and does not pass through any of the $c_i$.

To apply this lemma, let $a=f(x)$ and $b=f(y)$ (we may have already defined these values of $f$; if not, define them arbitrarily to be some points not in the image of $f$), let the $g_i$ be the smooth paths which we have already defined to be in the image of $f$ (there is one such path from each previous stage, so by CH, there are only countably many such paths), and let the $c_i$ be the other various points we have already defined to be in the image of $f$ (there are finitely many such points from each previous stage, so countably many in total by CH).  This gives us a smooth path from $a$ to $b$ which we can make the image of $f(A\cap S_\alpha)$ (except for the countably many points of $A\cap S_\alpha$ where we may have already defined $f$).  In addition, let us define $f$ at a couple more points to make sure $u$ is in both the domain and image of $f$.
At the end of this induction, we will have a path-connected bijection $f:\mathbb{R}^n\to\mathbb{R}^n$.  It is easy to see that we can arrange for $f$ to be discontinuous (at the $\alpha$th stage, we are free to choose any bijection at all between $A\cap S_\alpha$ and our path).
It remains to prove the Lemma.  To do so, note that the space $P$ of all smooth paths from $a$ to $b$ is a complete metric space in the natural $C^\infty$ topology.  For each $i$, let $U_i$ be the set of smooth paths that do not pass through $c_i$.  For each $i$, let $V_i$ be the set of smooth paths that do not intersect $g_i$ unless $a$ and/or $b$ is in the image of $g_i$, in which case they intersect only at $a$ and/or $b$ and have different derivatives there.  Then by standard transversality theory (using the fact that $n>2$), each $U_i$ and $V_i$ is an open dense subset of $P$.  By the Baire category theorem, there is an element of $P$ that is every $C_i$ and $D_i$, and this is our desired path.
As a final note, I expect that CH is not actually necessary here.  However, I don't quite see how to prove a version of the Lemma that works for a collection of $<\mathfrak{c}$ paths, rather than just for a countable collection of paths.  This question on MO seems related to this issue.
