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.

Consider the function $g(x) = f(x) - x$. Then $$g(a) = f(a) - a \ge a - a = 0$$ while $$g(b) = f(b) - b \le b - b = 0$$ So by the intermediate value theorem, ...?
