finding $C^1$ path on an open and path connected set. Given an open and path connected set $U\subseteq \mathbb R^n$, is there a way to find a $C^1$ path between every $a,b\in U$? 
If so, is there a general proof of existence of such path?

Given any two points in your domain, pick a continuous path between them. Because the closed interval $I$ is compact, you can cover this path with finitely many balls; use this to construct a piecewise linear path between the two points. (With enough care in this step and the previous you can ensure that the path is actually injective.) Now smooth the corners.
