Find an example of a function $f:[a,b]\to R$ which is continuous, but not strictly increasing, such that no inverse function ${f}^{-1}$ satisfy the property of the Inverse Function Theorem.

Jaqueline Kirby
2022-06-30
Answered

Find an example of a function $f:[a,b]\to R$ which is continuous, but not strictly increasing, such that no inverse function ${f}^{-1}$ satisfy the property of the Inverse Function Theorem.

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

$f,{f}_{1},{f}_{2},\dots $ bijective function with inverse functions $g,{g}_{1},{g}_{2},\dots $

${f}_{n}\to f$ pointwise for $n\to \mathrm{\infty}$.

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

$f,{f}_{1},{f}_{2},\dots $ bijective function with inverse functions $g,{g}_{1},{g}_{2},\dots $

${f}_{n}\to f$ pointwise for $n\to \mathrm{\infty}$.

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