I'm trying to cut down the use of differential calculus in the demonstration of the inverse function theoreme as an excercice to make it a little clearer to me :
Let's consider an application $T : \mathbb R^n \longrightarrow \mathbb R^n$ such as $T:x \longmapsto x+\phi(x)$ where $\phi: \mathbb R^n \longrightarrow \mathbb R^n$ is $\frac{1}{2}$-Lipschitz continuous.
I'm trying to show that T is a bijective application and that it is also an homeomorphism. I tried using the Banach fixed point theorem but i can't seem to find the right function.
I think not really understanding the standard proof keeps me from getting this one.