# continuous function, point in set

Let $f: [a,b] \rightarrow [a,b]$ be a continuous function. Prove that there is at least one point $x \in [a,b]$ so that $f(x)=x$.
This looks a lot like Intermediate Value Theorem but with the condition that $f(x)=x$. The condition that $f(x)=x$ is tricky for me, that there is an $x \in [a,b]$ that satisfies this. Thanks for any help.
Suppose that $g(x)=f(x)-x$ is $g$ continuous? WHY?
Is this true, $g(a)\geqslant 0\;\&\; g(b)\leqslant 0$? If so, then WHY?