ABSTRACT
Questions about limits go back to the ancient Greeks. The Greeks really did not understand limits. The question of limits arose even more intensely in the development of calculus. Isaac Newton did not understand limits, and neither did Leibniz. It took the combined efforts of a number of nineteenth-century mathematical geniuses, including Cauchy, Riemann, Dirichlet, Weierstrass, and others, to finally nail down the concept of limit. This chapter treats some topics related to one's calculus class and uses the deep properties of the real numbers developed in this text to obtain important new insights. The concept of continuous function is intuitively appealing. Having the idea of limit under control enables us to give a precise and accurate definition of continuity. Then some results about continuous functions are proved and developed a cogent theory. Continuous functions are very natural objects to study. Continuous functions interact very naturally with open sets, with compact sets, and with other standard artifacts of this theory.
