Preserving Bijectivity under pointwise limit

Question

The following question I have tried to solve and have also tried to find reference for. A reference would be appreciated.

Suppose that $f_i, g_i :\mathbb{R^n} \to \mathbb{R^n}$ are continuous functions such that $f_i \circ g_i =g_i \circ f_i = \text{id}_\mathbb{R^n}$ for each $i \in \mathbb{N}$. Let $\Omega_{\{f_i\}}$ be the set of points for which $\{f_i\}_i$ converges pointwise to some function, say $f$. Similarly, let $\Omega_{\{g_i\}}$ be the set of points for which $\{g_i\}_i$ converges pointwise to some function, say $g$. Then show that $$f \circ g = \text{id}_{\Omega_{\{g_i\}}}$$ and $$g \circ f = \text{id}_{\Omega_{\{f_i\}}}$$ specifically that such compositions make sense (i.e. that $\Omega_{\{f_i\}} \subseteq g(\Omega_{\{g_i\}})$ and $\Omega_{\{g_i\}} \subseteq f(\Omega_{\{f_i\}})$.

Answer 1

Counterexample on $\mathbb R^1:$ For $m=1,2,\dots,$ define

$$f_m(x)= \begin{cases}x,& x\le 0\\x^m,& 0<x<1\\ x,& x\ge 1\\\end{cases}$$

Then each $f_m$ is a homeomorphism of $\mathbb R$ onto $\mathbb R.$ Set $g_m= (f_m)^{-1},$ i.e.,

$$g_m(x)= \begin{cases}x,& x\le 0\\x^{1/m},& 0<x<1\\ x,& x\ge 1\\\end{cases}$$

Check that $f_m$ converges pointwise everywhere to the function $f$ that is $0$ on $(0,1)$ and is the identity elsewhere. Similarly, $g_m$ converges pointwise everywhere to the function $g$ that is $1$ on $(0,1)$ and is the identity elsewhere. We have $f\circ g=1$ on $(0,1),$ and $g\circ f=0$ on $(0,1),$ showing the desired conclusion does not hold.

Answer 2

It is not true.

Let us first consider the following sequence of piecewise linear homeomorphisms on the unit interval $I = [0,1]$: $$h_i(t) = \begin{cases} \frac{t}{i} & t \le \frac{1}{2} \\ \frac{1}{2i}+2(1-\frac{1}{2i})(t-\frac{1}{2}) & t \ge \frac{1}{2} \end{cases} $$ The inverse homeomorphisms are given by $$h_i^{-1} = \begin{cases} ti & t \le \frac{1}{2i} \\ \frac{1}{2}+\frac{1}{2(1-\frac{1}{2i})}(t-\frac{1}{2i}) & t \ge \frac{1}{2i} \end{cases} $$ Obviosuly $(h_i)$ converges (even uniformly) to the continuous surjection $$h : I \to I, h(t) = \begin{cases} 0 & t \le \frac{1}{2} \\ 2(t-\frac{1}{2}) & t \ge \frac{1}{2} \end{cases} $$ The sequence $(h_i^{-1})$ also converges pointwise on $I$. Its limit is $$h^- : I \to I, h^-(t) = \begin{cases} 0 & t = 0 \\ \frac{1}{2}+\frac{1}{2}t & t > 0 \end{cases} $$ This is a non-continuous injection. We have $h \circ h^- = id$, but $h^- \circ h \ne id$.

To get examples on $\mathbb{R}^n$, define $$f_i(x) = \begin{cases} x & \lVert x \rVert \ge 1 \\ h_i(\lVert x \rVert)x & \lVert x \rVert \le 1 \end{cases} $$