Give an example of a function $f:\mathbb{R} \to \mathbb{R}$ that satisfies all three conditions

Question

Give an example of a function $f:\mathbb{R} \to \mathbb{R}$ that satisfies all three conditions

1. $f$ is bijective
2. $f'(0)=0$
3. the inverse function $f^{-1}$ is not continuous at $0$

Does $f(x)=x^3$ satisfy the all three conditions?

If not can any provide me an example?

Answer 1

No, the last condition fails. The inverse $f^{-1}(y)=\sqrt[3]y$ is continuous at zero.

What would work, then? We want things to be nice at $x=0$ but not nice at the point where $f(x)=0$. Therefore we don't want $f(0)=0$. Your function $x\mapsto x^3$ only fails the third condition. To remedy this, we change it to $x\mapsto x^3+1$.

Now all we have to do is to modify this function at $x=-1$ and maybe elsewhere far from $x=0$ to make the inverse discontinuous. This is easiest to do by swapping two values: $$ f(x) = \begin{cases} x^3+1,&x\neq\pm1\\ 0,&x=+1\\ 2,&x=-1 \end{cases} $$ Draw a picture, and it should become clear why this works.