If there is a continuous function between linear continua, then this function has a fixed point? Let $f:X\to X$ be a continuous map and $X$ be a linear continuum. Is it true that $f$ has a fixed point?
I think the answer is "yes" and here is my proof:
Assume to the contrary that for any $x\in X$, either $f(x)<x$ or $f(x)>x$. Then, $A=\{x: f(x)<x\}$ and $B=\{x: f(x)>x\}$ are disjoint and their union gives $X$. Now if we can show that both $A$ and $B$ are open we obtain a contradiction because $X$ is connected. 
How can we show that $A$ and $B$ are open in $X$?

The function $f(x)=x+1$ is a counterexample.  Here both sets $A$ and $B$ are open, but one of them is empty :-)
Brouwer fixed point theorem asserts that the closed ball has the property you are looking for: every continuous self-map will have a fixed point.  But the proof requires tools well beyond the general topological arguments you outlined.  The most straightforward proof passes via relative homology or homotopy, and exploits the nontriviality of certain homology (resp. homotopy) classes.
