Continuous Functions in Metric Spaces

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1 Continuous Functions in Metric Spaces Throughout this section let (X, d X ) and (Y, d Y ) be metric spaces. Definition: Let x X. A function f : X Y is continuous at x if for every sequence {x n } that converges to x, the sequence {f(x n )} converges to f(x). Definition: A function f : X Y is continuous if it is continuous at every point in X. Theorem: A function f : X Y is continuous at x if and only if for every ɛ > 0 there is a δ > 0 such that d X (x, x) < δ d Y (f(x), f(x)) < ɛ i.e., ɛ > 0 : δ > 0 : x B(x, δ) f(x) B(f(x), ɛ). ( ) Proof: ( :) Let ɛ > 0. Suppose, by way of contradiction, that there is no δ > 0 such that d X (x, x) < δ d Y (f(x), f(x)) < ɛ i.e., δ > 0 : x B(x, δ) for which f(x) B(f(x), ɛ). Then, in particular, for every n N, let 1 play the role of δ above: there is an x n n B(x, 1 ) n for which f(x n ) B(f(x), ɛ). We therefore have a sequence {x n } in X that converges to x but the sequence {f(x n )} does not converge to f(x), contradicting our assumption that f is continuous. ( :) Assume that ( ) holds, and let {x n } be a sequence that converges to x. In order to show that {f(x n )} converges to f(x), let ɛ > 0. According to ( ), there is a δ > 0 for which x B(x, δ) f(x) B(f(x), ɛ). Since {x n } x, we can choose n N such that n > n x n B(x, δ). But then n > n f(x n ) B(f(x), ɛ); i.e., {f(x n )} converges to f(x), and f is therefore continuous at x. Remark: For functions f from R n to R m this theorem says that f is continuous at x R n if and only if for every ɛ > 0 there is a δ > 0 such that x x < δ f(x) f(x) < ɛ. Theorem: A function f : X Y is continuous if and only if for every open set V in Y the inverse image f 1 (V ) is an open set in X. Proof: Exercise. An elementary consequence of the preceding theorem is its analogue in terms of closed sets: Theorem: A function f : X Y is continuous if and only if for every closed set S in Y the inverse image f 1 (S) is a closed set in X.

2 This gives us four equivalent definitions of a continuous function f from X to Y : For every sequence {x n } that converges to x, the sequence {f(x n )} converges to f(x). For every x X : ɛ > 0 : δ > 0 : x B(x, δ) f(x) B(f(x), ɛ). The inverse image of every open set in Y is an open set in X. The inverse image of every closed set in Y is a closed set in X. For real-valued functions there s an additional, more economical characterization of continuity (where R is of course assumed to have the metric defined by the absolute value): Theorem: A real-valued function f : X R is continuous if and only if, for every c R the sets {x X f(x) < c} and {x X f(x) > c} are both open sets in X. And of course we therefore also have the parallel characterization in terms of closed sets: Remark: A real-valued function f : X R is continuous if and only if, for every c R the sets {x X f(x) c} and {x X f(x) c} are both closed sets in X. Exercise: Provide a proof of the above theorem. The proof is a straightforward application of the following two propositions, which we haven t proved but which are easy to prove: Proposition: Every open set in R is a union of open intervals. Proposition: For any function f : X Y, any set A, and any collection {S α α A} of sets S α Y : f 1 ( α A S α ) = α A f 1 (S α ) and f 1 ( α A S α ) = α A f 1 (S α ). Remark: When the target space Y is actually a normed vector space, it s natural to define the sum and scalar multiple of continuous functions pointwise i.e., the functions f + g : X Y and αf : X Y are defined by x X : (f + g)(x) = f(x) + g(x) and x X : (αf)(x) = αf(x). Then the set C(X; Y ) of all continuous functions on X into Y, with these definitions of addition and scalar multiplication, is a vector space. Proof: Exercise. This requires showing that C(X; Y ) is closed under vector addition and scalar multiplication. This does not mean that C(X; Y ) is a closed set, but rather that if f and g are in C(X; Y ) and α R, then f + g and αf are in C(X; Y ) i.e., that the sum of continuous functions is a continuous function, and that a multiple of a continuous function is a continuous function. For real-valued functions (i.e., if Y = R), we can also define the product f g and (if x X : f(x) 0) the reciprocal 1/f of functions pointwise, and we can show that if f and g are continuous then so are fg and 1/f. 2

3 Remark: If X, Y, and Z are metric spaces, and if f : X Y and g : Y Z are continuous, then the composition f g : X Z is continuous. The Weierstrass Theorem In Euclidean space (i.e., R n with any norm) we say that a set is compact if it s both closed and bounded. One of the most important properties of continuous functions is that they preserve compactness i.e., if X is a compact subset of R n and if f : X R m is a continuous function, then the image of X, f(x), is a compact set in R m. This is the Weierstrass Theorem. In fact, the Weierstrass Theorem holds in general metric spaces: Weierstrass Theorem: If X is compact and f : X Y is continuous, then f(x) is a compact subset of Y. Corollary: If f : X R is a continuous real-valued function on a compact set, then f attains a maximum and a minimum on X. Instead of proving the Weierstrass Theorem here, we defer the proof until after we ve developed our next important concept, the Bolzano-Weierstrass (B-W) Property. There are two good reasons for waiting until then to do the proof: (1) we need the B-W Property in order to generalize the notion of a compact set from R n to general metric spaces, and (2) the theorem s proof is much easier using the B-W Property in the general setting than if we were to do it using the closed-and-bounded definition of compactness in Euclidean space. Application to Utility Theory and the Theory of Choice In Example 3 in the Binary Relations lecture notes, we had a real-valued function u : X R. We interpreted X as a set of alternatives from which a decision-maker chooses, and we interpreted u as a utility function describing the decision-maker s preference in the sense that she strictly prefers an alternative x to an alternative x if and only if u(x ) > u(x). Then, from the function u, we defined binary relations R and P on X (alternatively denoted and ) to describe the idea of preference not in terms of a function u, but simply as preference between pairs of alternatives: x R u x u(x ) u(x) and x P u x u(x ) > u(x), ( ) and we noted that R u will be a complete preorder and P u will be the associated strict preorder. In other words, any real-valued function u on X defines a complete preorder R u on X that is naturally interpreted as the decision-maker s preference among the alternatives in X. 3

4 Now note that if X is a metric space, and if the function u : X R is continuous, then for every x X, the weak upper- and lower-contour sets Rx and xr are both closed sets in X: Rx = {x X u(x) u(x)} and xr = {x X u(x) u(x)}. And of course every strict upper- and lower-contour set P x and xp is an open set in X. Because preorders whose contour sets are all open or all closed are naturally associated with continuous functions in this way, it s natural and useful! to define such preorders themselves as continuous: Definition: A complete preorder R on a metric space (X, d) is continuous if all of its upper- and lower-contour sets Rx and xr are closed sets. Remark: A complete preorder R on a metric space is continuous if and only if, for the associated strict preorder P, all the upper- and lower-contour sets P x and xp are open sets. What we ve established in the preceding paragraphs is that for any continuous real-valued function u : X R, the relation R u defined by ( ) is a continuous preorder on X. But as we said in the Binary Relations notes, it s actually preferences that we think are fundamental, not utility functions. So instead of starting with a function u as the representation of someone s preference and defining an associated preorder R u, or, what we really want to be able to do is to start with some given preference relation (a complete preorder R), and define a utility function that represents it in the sense of ( ) (recall the definition of a representation of R from the Binary Relations notes). The Representation Theorem uses the notion of a continuous preorder to provide a sufficient condition that ensures we can do this. Representation Theorem: If a relation R on a metric space (X, d) is complete, transitive, and continuous i.e., a complete and continuous preorder then it is representable. Moreover, it is representable by a continuous utility function. Proof: Debreu, on page 56, Proposition (1), gives a proof. Jehle & Reny, on page 120, Theorem 3.1, give a proof for complete preorders that are continuous and strictly increasing. Combining the Representation Theorem with the Weierstrass Theorem gives us the following result that provides a sufficient condition on a preference R (or ) that ensures that on any compact set of alternatives (such as a budget set with positive prices) there will always be an alternative that s best according to. Weierstrass Theorem for Preorders: If X is compact and is a complete and continuous preorder on X, then there exists an x X that is a maximum for i.e., an x that satisfies x x for every x X. 4

5 Exercise: The lexicographic preference on R 2 + is a complete preorder. Is it continuous? Does it have a maximum on every compact (i.e., closed and bounded) subset of R 2 +? Exercise: Suppose u : X R is a real-valued function on a metric space X, and suppose the associated preorder R u defined in ( ) is continuous. Is u continuous? 5