Analysis Question? Fixed Point?

How would you go about proving this assertion?

$f:A \to A$ has a fixed point iff the graph of f intersects the diagonal

Also, in class we've proven that given $a,b \in \mathbb{R}$ with $a < b$ and letting $f:[a,b] \to [a,b]$ be continuous, $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.

But does this hold for $f:(a,b) \to (a,b)$ and discontinuous functions?