In this final chapter I am going to introduce you to the notion of a continu-ous function of the real numbers. This is a topic that deserves a whole bookto itself, but I am giving you this little taster here for several reasons. First,this notion is the right setting in which to prove the existence of nth roots ofpositive real numbers, which I promised to do way back in Chapter 5. Second,one can quite quickly develop enough theory to prove a famous result calledthe Intermediate Value Theorem, which sheds quite a bit of light on the greatFundamental Theorem of Algebra, stated in Chapter 7. And last, the idea ofa continuous function provides yet another example of something natural andobvious-sounding being turned into rigorous mathematics, with fruitful results. The material in this chapter is perhaps at a higher level than most of the restof the book, but it will repay study and hopefully serve as a useful introductionto your next course in analysis.