Brouwer's theorem 2.0? [closed]

Question

Let $f\in C([0,1]^n,\mathbb R^n) $ with $[0,1]^n \subset f([0,1]^n)$

Is it true that $\exists x \in [0,1]^n, f(x) =x$?

Answer 1

This is a simple counterexample .

You use two disjoint parts of the square to cover the square.

In one dimension you can prove this using the intermediate value theorem.