Jaqueline Kirby

2022-06-30

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.

Harold Cantrell

Beginner2022-07-01Added 21 answers

Any constant function $f(x)=c$ will be an example. $f$ is continuously differentiable, but ${f}^{-1}$ can't be a function as you would have ${f}^{-1}(c)=x$ and ${f}^{-1}(c)={x}^{\prime}$ where $x\ne {x}^{\prime}$, thereby violating the definition of a function.

Damon Stokes

Beginner2022-07-02Added 6 answers

Consider the function $f(x)={x}^{2}\mathrm{sin}(\frac{1}{x})$ defined on $[0,1]$. Then you can see that ${f}^{\prime}(0)=1$ but ${f}^{\prime}$ fails to be continuously differentiable as ${f}^{\prime}$ is not continuous at $0$. And also you can see that there is no neighborhood of $0$ such that ${f}^{-1}$ exists

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