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

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.
