Let $(X, \phi)$ and $(Y, \sigma)$ be metric spaces, and let

  1. $f, f_1, f_2, \ldots$ bijective function with inverse functions $g, g_1, g_2, \ldots$

  2. $f_n \to f$ pointwise for $n \to \infty$.

And all involved functions are continuous. Does it hold that $g_n \to g$ pointwise for $n \to \infty$?


Partial answer:

If $y=f(x)$ and $y_n=f_n(x)$ then $$ |g(y)-g_n(y)|=|x-g_n(\;y-y_n+y_n\;)|=|g_n(y_n)-g_n(y_n + e_n)| $$ where $e_n=y-y_n \rightarrow 0$.

Therefore if $\{g_n\}$ is uniformly equicontinuous the answer is yes: $$ |g(y)-g_n(y)|\le \sigma(e_n) $$ where $\sigma$ is the modulus of continuity of the family $\{g_n\}$.


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