Condition of the Brouwer fixed-point theorem

Question

While I know it is true that:

Every continuous function from a nonempty convex compact subset $K$ of an Euclidean space to $K$ itself has a fixed point. (Wikipedia)

I am not sure whether a continuous function whose domain is $[0,1]$ but with the range being $(0,1)$ can still apply the theorem and prove the existence of a fixed point. Also, for those functions whose domain takes $[0,+\infty)$, can I use a sufficiently large number $M$ to show $[0,M]\rightarrow [0, M]$ to prove the existence of a fixed point between $[0, M]$?

Answer 1

1. For each continuous function $f : [0,1] \to (0,1)$ there exists a point $x_0 \in (0,1)$ such that $f(x_0) = x_0$.

To see this, let $i : (0,1) \to [0,1]$ denote the inclusion function. Then $F = i \circ f : [0,1] \to [0,1]$ is continuous and has a fixed point $x_0 \in [0,1]$. We get $f(x_0) = F(x_0) = x_0$ and $x_0 = f(x_0) \in (0,1)$.

2. It is not clear which functions you consider on $[0,\infty)$.

If you are interested in functions $f : [0,\infty) \to [0,\infty)$, the answer is "no". As an example take $f(x) = x + 1$.

If you are interested in functions $f : [0,\infty) \to [0,M]$, the answer is "yes". Just take the restriction $f \mid_{[0,M]}$ which has a fixed point.