Brouwer's fixed point theorem states that any continuous function $f$ mapping a compact convex set into itself has a fixed point, i.e. a point such that $f(x)=x$.
I am somewhat unclear on the restriction on the domain and range. Suppose the domain of the function is a compact and convex set $A$. I understand the theorem to require $f$ to be a mapping $A \to A$. Must the function $f$, however, be able to take on all values in $A$?
Suppose we certainly know that the range of the continuous function $f(x)$ is within the set $B \subset A$. Does Brouwer's fixed point theorem still apply? I would answer yes, beacause we can still state that the range of $f$ is the set $A$, but am not sure.
In two dimensions, I can show using the Intermediate Value Theorem, that in such a case there is a fixed point. For example if $x \in [0,a]$ and I know that $f(x) \in [1,b]$ (specifically, $f(0)=1$ and$f(a)=b$), where $a>1$ and $b<a$, then by the intermediate value theorem, it follows that a fixed point exists.
I would also conclude, that if insteaed I know that in this case, if $f(x)\in[-1,b]$, that the Brouwer fixed point theorem does not apply. Is that correct?