This question seems fairly logical if you think about it as the two functions must intersect. How would you go about proving it?

$\displaystyle Suppose\ that\ f\ and\ g\ are\ continuous$ $\displaystyle functions\ on\ the\ closed\ interval\ [a,b]$. $\displaystyle Show\ that\ there\ exists\ a\ point\ c \in [a,b]$ $\displaystyle such\ that\ f(c)=g(c)\ if\ f(a) \leq g(a)\ and\ f(b) \geq g(b)$