MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 1 [made Community Wiki]

The following problem is often found in introductory Real Analysis courses but can be solved by IVT:

Let $f :[0,1] \to [0,1]$ be continuous. Show that f(x) has a fixed point. In other words, there exists

$y \in [0,1]$ such that $f(y) = y$.