Find an example of continuous but not increasing function whose inverse function doesn't satisfy the Inverse Function Theorem

Question

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

Answer 1

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 $f^{-1}(c)=x'$ where $x \neq x'$, thereby violating the definition of a function.

Answer 2

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