Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$ - that is, $x$ such that $f(x) = x$. Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$ -  that is, $x$ such that $f(x) = x$.
I haven't a clue where to even start on this one.

If $f(a)=a$ or $f(b)=b$ then we are done. Otherwise assume that $f(a)>a$ and $f(b)<b$. Consider $g(x)=f(x)-x$. $g(a)>0$ but $g(b)<0$ hence there exists $c\in(a,b)$ such that $g(c)=0$. Hence $f(c)=c$.
