Download Continuous Functions in Metric Spaces

Continuous Functions in Metric Spaces
Throughout this section let (X, dX ) and (Y, dY ) be metric spaces.
Definition: Let x ∈ X. A function f : X → Y is continuous at x if for every sequence {xn }
that converges to x, the sequence {f (xn )} 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 > 0 there is a
δ > 0 such that dX (x, x) < δ ⇒ dY (f (x), f (x)) < — i.e.,
∀ > 0 : ∃δ > 0 : x ∈ B(x, δ) ⇒ f (x) ∈ B(f (x), ).
(∗)
Proof:
(⇒:) Let > 0. Suppose, by way of contradiction, that there is no δ > 0 such that
dX (x, x) < δ ⇒ dY (f (x), f (x)) < — i.e.,
∀δ > 0 : ∃x ∈ B(x, δ) for which f (x) 6∈ B(f (x), ).
Then, in particular, for every n ∈ N, let
1
n
play the role of δ above: there is an xn ∈ B(x, n1 )
for which f (xn ) 6∈ B(f (x), ). We therefore have a sequence {xn } in X that converges to x
but the sequence {f (xn )} does not converge to f (x), contradicting our assumption that f is
continuous.
(⇐:) Assume that (∗) holds, and let {xn } be a sequence that converges to x. In order to
show that {f (xn )} converges to f (x), let > 0. According to (∗), there is a δ > 0 for which
x ∈ B(x, δ) ⇒ f (x) ∈ B(f (x), ).
Since {xn } → x, we can choose n ∈ N such that n > n ⇒ xn ∈ B(x, δ). But then
n > n ⇒ f (xn ) ∈ B(f (x), );
i.e., {f (xn )} converges to f (x), and f is therefore continuous at x. k
Remark: For functions f from Rn to Rm this theorem says that f is continuous at x ∈ Rn
if and only if for every > 0 there is a δ > 0 such that kx − xk < δ ⇒ kf (x) − f (x)k < .
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.
This gives us four equivalent definitions of a continuous function f from X to Y :
For every sequence {xn } that converges to x, the sequence {f (xn )} converges to f (x).
For every x ∈ X : ∀ > 0 : ∃δ > 0 : x ∈ B(x, δ) ⇒ f (x) ∈ B(f (x), ).
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) 5 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 {Sα | α ∈ A} of
sets Sα ⊆ Y :
f −1 (∪α∈A Sα ) = ∪α∈A f −1 (Sα )
and
f −1 (∩α∈A Sα ) = ∩α∈A f −1 (Sα ).
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) 6= 0) the reciprocal 1/f of functions pointwise, and we can show that if f
and g are continuous then so are f g and 1/f .
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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., Rn 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 Rn and if f : X → Rm
is a continuous function, then the image of X, f (X), is a compact set in Rm . 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 Rn 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 x0 to an alternative x if and only if u(x0 ) > 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:
x0 Ru x ⇔ u(x0 ) = u(x) and x0 Pu x ⇔ u(x0 ) > u(x),
(∗∗)
and we noted that Ru will be a complete preorder and Pu will be the associated strict preorder.
In other words, any real-valued function u on X defines a complete preorder Ru on X that
is naturally interpreted as the decision-maker’s preference among the alternatives in X.
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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) 5 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 Ru 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 Ru , 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
b ∈ X that is a maximum for % — i.e., an x
b that satisfies
x
b % x for every x ∈ X.
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Exercise: The lexicographic preference on R2+ is a complete preorder. Is it continuous?
Does it have a maximum on every compact (i.e., closed and bounded) subset of R2+ ?
Exercise: Suppose u : X → R is a real-valued function on a metric space X, and suppose
the associated preorder Ru defined in (∗∗) is continuous. Is u continuous?
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