I was wondering whether there are any theorems on the closure w.r.t. the $L^p$-norm, $p\in[1,\infty]$, of the set of continuous bijective functions with continuous inverse defined on a compact subset $K$ of $\mathbb{R}^d$.
Any type of information would be helpful, e.g. what certainly does not lie in the closure, necessary conditions for the limits of such functions,...