# Convergence of injective functions

On the complex plane, it is true that if $\{ f_n\}$ is a set of holomorphic injective functions on the complex plane defined on a connected open set $\Omega$, which are convergent to $f$ uniformly on compact subsets of $\Omega$,then either $f$ is a constant function or is injective on $\Omega$ (of course, it is also holomorphic). The proof of the above fact is a few lines with the knowledge of Hurwitz's theorem.

I'd like to ask if this is true for injective functions on $\mathbb{R}$. So for example, does there exist a connected open set in $\mathbb{R}$, and $C^\infty$ functions $\{f_n\}$ on it which are injective, such that they converge to a function $f$ on the standard sup norm metric which is not injective or constant? This would characterize another difference between the complex and real numbers in a comprehensive manner.

The corresponding statement is not true for $C^\infty$ function on $\Bbb R$:
Let $f: \Bbb R \to \Bbb R$ be any (weakly) increasing $C^\infty$ which is not injective, for example $$f(x) = \begin{cases} 0 & \text{ for } x \le 0 \\ e^{-1/x} & \text{ for } x > 0 \end{cases}$$ Then the functions $$f_n(x) = f(x) + \frac 1n \arctan x$$ are strictly increasing (and therefore injective), and $f_n \to f$ uniformly on $\Bbb R$.