Test 2 Flashcards
Cantor Set
a set Cn consisting of 2^n closed intervals each having length 1/(3^&n). Finally, we define the Cantor set C (Fig. 3.1) to be the intersection of all the Cns
Properties:
C is uncountable
C contains only the endpoints of each sub interval
C is a perfect set
open set
A set O ⊆ R is open if for all points a ∈ O there exists an ε-neighborhood Vε(a) ⊆ O
limit point
A point x is a limit point of a set A if every ε-neighborhood Vε(x) of x intersects the set A at some point other than x.
x is quite literally the limit of a sequence in A
isolated point
A point a ∈ A is an isolated point of A if it is not a limit point of A
closed set
A set F ⊆ R is closed if it contains its limit points
closure
Given a set A ⊆ R, let L be the set of all limit points of A. The closure of A is defined to be A-closure = A ∪ L.
complement
Ac = {x ∈ R : x /∈ A}.
A set O is open if and only if Oc is closed. Likewise, a set F is closed if and only if Fc is open
compact set
A set K ⊆ R is compact if and only if it is closed and bounded.
open cover
An open cover for A is a (possibly infinite) collection of open sets {Oλ : λ ∈ Λ} whose union contains the set A; that is, A ⊆ UOλ.
Given an open cover for A, a finite subcover is a finite subcollection of open sets from the original open cover whose union still manages to completely contain A.
A set is compact if every open cover of A contains a finite subcover of A
perfect set
A set P ⊆ R is perfect if it is closed and contains no isolated points.
Closed intervals (other than the singleton sets [a, a]) serve as the most
obvious class of perfect sets, but there are more interesting examples.
separated sets
Two nonempty sets A, B ⊆ R are separated if A-closure ∩ B and
A ∩ B-closure are both empty
connected sets
A set that is not disconnected is called a connected set.
disconnected set
A set E ⊆ R is disconnected if it can be written as E = A ∪ B, where A and B are nonempty separated sets.
Fσ sets
A set A ⊆ R is called an Fσ set if it can be written as the countable union of closed sets
Gd set
A set B ⊆ R is called a Gd set if it can be
written as the countable intersection of open sets.
nowhere-dense sets
A set E is nowhere-dense if E-closure contains no nonempty open intervals.
limit of a function
Let f : A → R, and let c be a limit
point of the domain A. We say that limx→c f(x) = L provided that, for all
ε > 0, there exists a δ > 0 such that whenever 0 < |x − c| < δ (and x ∈ A) it
follows that |f(x) − L| < ε
continuous
A function f : A → R is continuous at a
point c ∈ A if, for all ε > 0, there exists a
δ > 0 such that whenever |x − c| < δ
(and x ∈ A) it follows that
|f(x) − f(c)| < ε
If f is continuous at every point in the domain A, then we say that f is continuous on A
uniform continuity
A function f : A → R is uniformly
continuous on A if for every ε > 0 there exists a δ > 0 such that for all x, y ∈ A,
|x − y| < δ implies |f(x) − f(y)| < ε
intermediate value property
A function f has the intermediate value property on an interval [a, b] if for all
x < y in [a, b] and all L between f(x) and f(y), it is always possible to find a point c ∈ (x, y) where f(c) = L
sets of discontinuity (Df)
Given a function f : R → R, define Df ⊆ R to be the set of points where the function f fails to be continuous
monotone, increasing, decreasing functions
A function f : A → R is increasing on A if f(x) ≤ f(y) whenever x < y
decreasing if f(x) ≥ f(y) whenever x < y
A monotone function is one that is either increasing or decreasing.
removable discontinuity
If limx→c f(x) exists but has a value different from f(c), the discontinuity
at c is called removable
jump discontinuity
If limx→c+ f(x) =/= limx→c− f(x),
then f has a jump discontinuity at c
essential discontinuity
If limx→c f(x) does not exist for some other reason, then the discontinuity
at c is called an essential discontinuity.
α-continuous
Let f be defined on R, and let α > 0. The function f is α-continuous at x ∈ R if there exists a δ > 0 such that for all y, z ∈ (x−δ, x+δ) it follows that |f(y) − f(z)| < α
Df-α
Df-alpha = {x ∈ R : f is not α-continuous at x}.
Characterization of Compactness in R
A set K ⊆ R is compact if and only if it is closed and bounded
Nested Compact Set Property
If K1 ⊇ K2 ⊇ K3 ⊇ K4 ⊇ …
is a nested sequence of nonempty compact sets, then the infinite intersection Kn is not empty.
A nonempty perfect set is
uncountable
Sequential Criterion for Functional Limits
Given a function f : A → R and a limit point c of A, the following two statements are
equivalent:
(i) lim (x→c) f(x) = L
(ii) For all sequences (xn) ⊆ A satisfying xn =/= c and (xn) → c, it follows that f(xn) → L.
Characterization of Continuity
The function f is continuous at c if and only if any one of the following
three conditions is met:
(i) For all ε > 0, there exists a δ > 0 such that |x−c| < δ (and x ∈ A) implies
|f(x) − f(c)| < ε
(ii) For all V(f(c)), there exists a Vδ(c) with the property that x ∈ Vδ(c) (and x ∈ A) implies f(x) ∈ Vε(f(c))
(iii) For all (xn) → c (with xn ∈ A), it follows that f(xn) → f(c)
If c is a limit point of A, then the above conditions are equivalent to
(iv) lim x→c f(x) = f(c).
Algebraic Limit Theorem for Functional Limits
(i) lim x→c kf(x) = kL for all k ∈ R,
(ii) lim x→c [f(x) + g(x)] = L + M,
(iii) limx→c [f(x)g(x)] = LM
(iv) lim x→c f(x)/g(x) = L/M, M =/= 0
Divergence criterion for functional Limits
Let f be a function defined on A, and let c be a limit point of A. If there exist two
sequences (xn) and (yn) in A with
xn =/= c and yn =/= c and
lim xn = lim yn = c
but lim f(xn) =/= lim f(yn),
then we can conclude that the functional limit lim x→c f(x) does not exist
Criterion for Discontinuity
Let f : A → R, and let c ∈ A be a limit point of A. If there exists a sequence (xn) ⊆ A where (xn) → c but such that f(xn) does not converge to f(c), we may conclude that f is not continuous at c.
Algebraic Continuity Theorem
Assume f : A → R and g : A → R are continuous at a point c ∈ A. Then,
(i) kf(x) is continuous at c for all k ∈ R;
(ii) f(x) + g(x) is continuous at c;
(iii) f(x)g(x) is continuous at c; and
(iv) f(x)/g(x) is continuous at c, provided the quotient is defined.
Composition of Continuous
Functions
Given f : A→R and g : B → R, assume that the range f(A) = {f(x) : x ∈ A} is contained in the domain B so that the composition g ◦ f(x) = g(f(x)) is defined on A. If f is continuous at c ∈ A, and if g is continuous at f(c) ∈ B, then g ◦ f is
continuous at c.
Preservation of Compact Sets
Let f : A → R be continuous on A. If K ⊆ A is compact, then f(K) is compact as well.
Extreme Value Theorem
If f : K → R is continuous on
a compact set K ⊆ R, then f attains a maximum and minimum value.
In other words, there exist x0, x1 ∈ K such that f(x0) ≤ f(x) ≤ f(x1) for all x ∈ K.
Sequential Criterion for Absence of Uniform Continuity
A function f : A → R fails to be uniformly continuous on A if and only if there exists a particular ε0 > 0 and two sequences (xn) and (yn) in A satisfying
|xn − yn| → 0 but |f(xn) − f(yn)| ≥ ε0.
A set containing only isolated points must be closed
False
The set Z is a nowhere-dense set
True
Let A be a set and c element of A be an isolated point. Then
lim x->c f(x) = f(a)
False
The function f(x) = x^2 is uniformly continuous on [0, 1)
False
Suppose A is an open set. Then A-closure is perfect
True
Let f(x) : R -> R be a continuous function, and suppose for any x except possibly 1, f(x) > 0. Then, f(1) >= 0
True
Let f : R ! R be a function. If x ∈ Df , then x ∈ Df-a for every alpha.
False
Let (Ai) be a countable sequence of open sets. Then
The closure of the union of all Ai is equal to the union of all Ai closures
False
If A is open and unbounded, then Ac is compact
False
