If I consider the inverse function theorem for Banach spaces, I'd like to know if there is an counter-example where I could weaken my space to a non-complete space, like: $\mathbb{R}[X]$ for any norm.
I could use $(C[0, 1], \Bbb R) \to (C[0, 1], \Bbb R)$ using different norms and the identity as a counter example.
But I'd like to have the same space using the same norm.
Bonus if the inverse of the function is not continuous.
I know that Inverse Operator Theorem, counter example gives a counter-example, but it's using different norms.