Consider a vector-valued function $f: [0,1]^n\rightarrow[0,1]^n$. Write $f(x)=\{f\_1(x), ..., f\_n(x)\}$ with $x\in[0,1]^n$, where the $f\_i: [0,1]^n\rightarrow[0,1]$ are continuous functions with the following properties:


1) $f\_i(\{x\_1, x\_2, ..., x\_n\})=0$ if $x\_i=0$.


2) $f\_i(\{x\_1, x\_2, ..., x\_n\})=1$ if $x\_i=1$.


Fix $p\in[0,1]^n$. I want to prove that there exists a solution $x\in[0,1]^n$, such that $f(x)=p$.


If feel that this should be true, since by the intermediate value theorem, we know that for fixed $x\_2$, ..., $x\_n$ we can find $x\_1$ such that $f\_1(\{x\_1, x\_2, ..., x\_n\})=p\_1$ (and similarly for $f\_2$ we can find an $x\_2$ if we fix the other $x\_i$'s).


Any help is appreciated. 



Intermediate value for a vector-valued function
This has its own [Wikipedia entry Poincaré–Miranda theorem](https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Miranda_theorem) and is similar to the Brouwer fixed-point theorem.


