Chapter D Continuity I This chapter provides a basic introduction to the theory of functions in general, and to that of continuous maps between two metric spaces in particular. Many of the results that you have seen in your earlier studies in terms of real functions on R are derived here in the context of metric spaces. | Chapter D Continuity I This chapter provides a basic introduction to the theory of functions in general and to that of continuous maps between two metric spaces in particular. Many of the results that you have seen in your earlier studies in terms of real functions on R are derived here in the context of metric spaces. Examples include the Intermediate Value Theorem Weierstrass Theorem and the basic results on uniform convergence such as those about the interchangeability of limits and Dini s Theorem . We also introduce and lay out a basic analysis of a few concepts that may be new to you like stronger notions of continuity . uniform Lipschitz and Holder continuity weaker notions of continuity . upper and lower semicontinuity homeomorphisms and isometries. This chapter contains at least four topics that are often not covered in standard courses on real analysis but that nevertheless see good playing time in various branches of economic theory. In particular and as applications of the main body of the chapter we study Caristi s famous generalization of the Banach Fixed Point Theorem the characterization of additive continuous maps on Euclidean spaces and de Finetti s theorem on the representation of additive preorders. We also revisit the problem of representing a preference relation by a utility function and discuss the two of the best known results of utility theory namely the Debreu and Rader Utility Representation Theorems. This completes our coverage of ordinal utility theory we will be able to take up issues related to cardinal utility only in the second part of the text. The pace of the chapter is leisurely for the most part and our treatment is fairly elementary. Towards the end however we study two topics that may be considered relatively advanced. These may be omitted in the first reading. First we discuss Marshall Stone s important generalization of the Weierstrass Approximation Theorem. We prove this landmark result and consider a few of its .
