§2.1–Metric Space Concepts, Part A
Tom Lewis
Fall Term 2006
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
1 / 12
Outline
1
Measuring distance
2
Convergent sequences and subsequences
3
Continuity
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
2 / 12
Measuring distance
Definition
A metric space is a set M endowed with a function d : M × M → R
satisfying
d(x, y ) ≥ 0 for all x, y ∈ M and d(x, y ) = 0 if and only if x = y .
d(x, y ) = d(y , x) for all x, y ∈ M;
d(x, y ) ≤ d(x, z) + d(z, y ) for all x, y , z ∈ M.
The function d is called a metric. We often call d(x, y ) the distance from
x to y . The pair (M, d) is called a metric space
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
3 / 12
Measuring distance
Example
R2 endowed with the customary Euclidean metric
q
d(x, y ) = |x − y | = (x1 − y1 )2 + (x2 − y2 )2
is a metric space.
Rn endowed with the customary Euclidean metric
d(x, y ) = |x − y |
is a metric space.
R2 endowed with the taxi-cab metric
dT (x, y ) = dT ((x1 , x2 ), (y1 , y2 )) = |x1 − y1 | + |x2 − y2 |
is a metric space.
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
4 / 12
Measuring distance
Example
Given any set M, define d : M × M → R as follows:
(
1 if x 6= y
d(x, y ) =
0 if x = y
d is a metric, called the discrete metric. It is a useful source for
counterexamples.
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
5 / 12
Convergent sequences and subsequences
Definition
Let (M, d) be a metric space. A sequence (pn ) of elements of M
converges to a limit p ∈ M provided that for all ε > 0 there exists an
N ∈ N such that n ≥ N implies
d(pn , p) < ε.
We will write either
lim pn = p,
n→∞
pn →
p as n → ∞, or
pn → p.
to express the fact that (pn ) converges to p.
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
6 / 12
Convergent sequences and subsequences
Problem
Show that the sequence (pn ) with
2n + 1 1
, 2
pn =
n
n
converges in R2 to p = (2, 0).
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
7 / 12
Convergent sequences and subsequences
Example
Let pn = 1/n for n ≥ 1. The sequence of even numbered terms
1/2,
1/4,
1/6, . . .
and the sequence of prime-numbered terms
1/2,
1/3,
1/5,
1/7,
1/11,
···
are examples of subsequences of (pn ).
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
8 / 12
Convergent sequences and subsequences
Definition
Let (pn , ≥ 1) be a sequence and let 1 ≤ n1 < n2 < n3 < · · · be an infinite,
ordered collection of natural numbers. The sequence (qk , k ≥ 1) given by
qk = pnk
is called a subsequence of (pn ).
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
9 / 12
Convergent sequences and subsequences
Definition
Let (pn , ≥ 1) be a sequence and let 1 ≤ n1 < n2 < n3 < · · · be an infinite,
ordered collection of natural numbers. The sequence (qk , k ≥ 1) given by
qk = pnk
is called a subsequence of (pn ).
Theorem
Every subsequence of a convergent sequence converges and converges to
the same limit as the mother sequence.
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
9 / 12
Continuity
Continuity in R
Let A ⊂ R and let f : A → R be a function. We say that f is continuous
at x ∈ A if for every ε > 0 there exists a δ > 0 such that
|x − y | < δ
Tom Lewis ()
implies
|f (x) − f (y )| < ε
§2.1–Metric Space Concepts, Part A
Fall Term 2006
10 / 12
Continuity
Continuity in R
Let A ⊂ R and let f : A → R be a function. We say that f is continuous
at x ∈ A if for every ε > 0 there exists a δ > 0 such that
|x − y | < δ
implies
|f (x) − f (y )| < ε
Problem
Let f : (0, +∞) → R according to the rule f (x) = 1/x. Show that f is
continuous at x = 3.
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
10 / 12
Continuity
Continuity in R
Let A ⊂ R and let f : A → R be a function. We say that f is continuous
at x ∈ A if for every ε > 0 there exists a δ > 0 such that
|x − y | < δ
implies
|f (x) − f (y )| < ε
Problem
Let f : (0, +∞) → R according to the rule f (x) = 1/x. Show that f is
continuous at x = 3.
Definition
Let A ⊂ R and let f : A → R be a function. We say that f is continuous
(on A) if f is continuous at each point of A.
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
10 / 12
Continuity
Definition
Let (M, dM ) and (N, dN ) be metric spaces and let f : M → N be a
function. We say that f is continuous at x ∈ M if for every ε > 0 there
exists a δ > 0 such that
y ∈ M and dM (x, y ) < δ
implies
dN (f (x), f (y )) < ε
f is said to be continuous (on M) provided that f is continuous at each
point of M.
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
11 / 12
Continuity
Theorem
Let (M, dM ) and (N, dN ) be metric spaces. f : M → N is continuous if
and only if it sends each convergent sequence in M to a convergent
sequence in N, limits being sent to limits.
Theorem
Let U, V , and W be metric spaces and let f : U → V and g : V → W be
continuous functions. The composite function g ◦ f : U → W is
continuous.
Comment
In short, the composite of continuous functions is continuous.
Tom Lewis ()
§2.1–Metric Space Concepts, Part A
Fall Term 2006
12 / 12