Intro To Real Analysis Test 2

Front 1

Divergence Criterion for Limits

Back 1

Let f:A->R and let c be a cluster point of A. The following are equivalent:
-IF L∈R, then f doesn not have limit L at c iff there exists a sequence xₒ in A with xₒ≠c for all n∈N s.t. the sequence xₒ converges to c but the sequence (f(xₒ)) does not converge to L.
-The function f does not have a limit at c iff there exists a sequence xₒ in A with xₒ≠c for all n∈N s.t. the sequence xₒ converges to c but the sequence (f(xₒ)) does not converge in R.

Front 2

Definition of continuous function

Back 2

Let A⊆R, let f:A->R, and let c∈A. We say that f is continuous at c if, given any number ε>0 there exists δ>0 s.t. if x is any point of A satisfying ∣x-c∣<δ, then ∣f(x)-f(c)∣<ε.

Front 3

Discontinuity Criterion

Back 3

Let A⊆R, let f:A->R, and let c∈A. Then f is discontinuous at c iff there exists a sequence (xₒ) in A s.t. (xₒ) converges to c, but the sequence f(xₒ) does not converge to f(c).

Front 4

Definition of continuous set

Back 4

Let A⊆R, let f:A->R. If B is a subset of A, we say that f is continuous on the set B if f is continuous at every point of B.

Front 5

Sequential Criterion for limits

Back 5

Let f:A->R and let c be a cluster point of A. Then the following are equivalent.
1)lim f as x approaches c = L
2)for every sequence (xₒ) in A that converges to c s.t. xₒ≠c for all n∈N, the sequence (f(xₒ)) converges to L.

Front 6

Definition of the derivative

Back 6

Let I⊆R and f:I->R and c∈I. A real number L is the derivative of f at c if for any ε>0 there exists δ(ε)>0 s.t. if x∈I satisfies 0<∣x-c∣<δ(ε), then ∣[f(x)-f(c)/x-c]-L∣<ε.

Front 7

Caratheodory's Theorem

Back 7

Let f be defined on an interval I containing the point c. Then f is differentiable at c iff there exists a function φ on I that is continuous at c and satisfies
f(x)-f(c)=φ(x)(x-c) for x∈I.

Front 8

Relative maximum

Back 8

The function f:I→R has a relative maximum at c∈I if there exists a neighborhood V:=Vδ(c) of c such that f(x)≤f(c) for all x in V⋂I.

Front 9

Relative minimum

Back 9

The function f:I→R has a relative minimum at c∈I if there exists a neighborhood V:=Vδ(c) of c such that f(c)≤f(x) for all x in V⋂I.

Front 10

Interior Extremum Theorem

Back 10

Let c be an interior point of the interval I at which f:I→R has a relative extremum. If the derivative of f at c exists, then f'(c)=0.

Front 11

Rolle's Theorem

Back 11

Suppose that f is continuous on a closed interval I:=[a,b], that derivative f' exists at every point of the open interval (a,b), and that f(a)=f(b)=0. Then there exists at least one point c in (a,b) s.t. f'(c)=0.

Front 12

Mean Value Theorem

Back 12

Suppose that f is continuous on a closed interval I:=[a,b], and that f has a derivative in the open interval (a,b). Then there exists at least one point c in (a,b) s.t. f(b)-f(a)=f'(c)(b-a).

Front 13

Definition of Cauchy Sequence

Back 13

A sequence X=(xₒ) of real numbers is a Cauchy Sequence if for every ε>0 there exists a natural number N such that for all natural numbers n,m≥N ∣x(n)-x(m)∣<ε

Front 14

Definition of Cluster Point

Back 14

Let A⊆R. A point c∈R is a cluster point of A if for every δ>0 there exists at least one point x∈A, x≠c s.t. ∣x-c∣<δ

Front 15

Definition of a limit

Back 15

Let A⊆R and let a point c∈R be a cluster point of A. for a function f:A→R, a real number L is said to be a limit of f at c if, given any ε>0 there exists δ>0 s.t. if x∈A and 0<∣x-c∣<δ, then ∣f(x)-L∣<ε

Front 16

Definition of right-handed limit

Back 16

Let A⊆R and let f:A→R.
-If c∈R is a cluster point of A⊓(c,∞)={x∈A:x>c}, then L is a right-handed limit of f at c if, given any ε>0 there exists δ=δ(ε)>0 s.t. for all x∈A with 0<x-c<δ, then ∣f(x)-L∣<ε

Front 17

Definition of left-handed limit

Back 17

Let A⊆R and let f:A→R.
-If c∈R is a cluster point of A⊓(-∞,c)={x∈A:x<c}, then L is a left-handed limit of f at c if, given any ε>0 there exists δ>0 s.t. for all x∈A with 0<c-x<δ, then ∣f(x)-L∣<ε

Front 18

Definition of Infinite limits ∞

Back 18

Let A⊆R and let f:A→R and let c∈R be a cluster point of A.
-f tends to ∞ as x→c, if for every λ∈R there exists δ=δ(λ)>0 s.t. for all x∈A with 0<∣x-c∣<δ, then f(x)>λ

Front 19

Definition of Infinite limits -∞

Back 19

Let A⊆R and let f:A→R and let c∈R be a cluster point of A.
-f tends to -∞ as x→c, if for every λ∈R there exists δ=δ(λ)>0 s.t. for all x∈A with 0<∣x-c∣<δ, then f(x)<λ

Front 20

Definition of Limits at Infinity

Back 20

Let A⊆R and let f:A→R.
Suppose (a,∞)⊆A for some a∈A.
Real number L is a limit of f as x→∞ if given any ε>0 there exists K(ε)>a s.t. for any x>K, then ∣f(x)-L∣<ε

Front 21

Definition of continuous function at c

Back 21

Let A⊆R, let f:A→R, and let c∈A.
f is continuous at c if, given any ε>0 there exists δ>0 s.t. if x∈A satisfying ∣x-c∣<δ then ∣f(x)-f(c)∣<ε

Front 22

Definition of continuous function on a set

Back 22

Let A⊆R, let f:A→R.
If B is a subset of A, f is continuous on the set B if f is continuous at every point of B.

Front 23

Definition of uniformly continuous

Back 23

Let A⊆R, let f:A→R.
f is uniformly continuous on A if for each ε>0 there exists δ(ε)>0 s.t. if x,u∈A satisfying ∣x-c∣<δ(ε) then ∣f(x)-f(u)∣<ε

Front 24

Definition of Lipschitz function

Back 24

Let A⊆R, let f:A→R.
If there exists a constant K>0 s.t. ∣f(x)-f(u)∣≤K∣x-u∣ for all x,u∈A then f is a Lipschitz function.

Front 25

Open Set

Back 25

A subset G of R is open in R if for each x∈G there exists a neighborhood V of x s.t. V⊆G.

Front 26

Closed Set

Back 26

A subset G of R is closed in R if the complement C(F):=R\G is open in R.

Front 27

Definition of a partition

Back 27

A partition of an interval I:=[a,b] is a collection P={I1,....In} of non-overlapping closed intervals whose union is [a,b].

Front 28

Definition of a tagged partition

Back 28

If a point t(i) has been chosen from each Interval I(i) then the points t(i) are called the tags and the set of ordered pairs P={(I1,t1),....(I(n),t(n))) is called a tagged partition of I.

Front 29

Definition of a guage

Back 29

A guage on I is a strictly positive function defined on I.

Front 30

Definition of δ-fine

Back 30

If δ is a gauge on I, then a partition P is said to be δ-fine if t(i)∈I(i)⊆[t(i)-δt(i),t(i)+δt(i)]

Front 31

Definition of Riemann Integral

Back 31

A function f:[a,b]→R is Riemann integrable on [a,b] if there exists a number L∈R s.t. for every ε>0 there exists δ(ε)>0 s.t. if P is any tagged partition of [a,b] with norm of P<δ(ε), then ∣S(f:P)-L∣<ε

Front 32

Definition of Riemann sum

Back 32

Define Riemann sum of a function f:[a,b]→R corresponding to a tagged partition P to be the number
S(f:P):=∑f(t(i)(x(i)-x(i-1))

Front 33

Cauchy Convergence Criterion

Back 33

A sequence of real numbers is convergent iff it is a Cauchy sequence.

Front 34

Limit Squeeze Theorem

Back 34

Let A⊆R, let f,g,h:A->R, and let c∈R be a cluster point of A. If
f(x)≤g(x)≤h(x) for all x∈A,x≠c and if lim f=L=lim h, then lim g=L.

Front 35

Maximum-Minimum Theorem

Back 35

Let I:=[a,b] be a closed bounded interval and let f:I→R be continuous on I. Then f has an absolute maximum and an absolute minimum on I.

Front 36

Bolzano's Intermediate Value Theorem

Back 36

Let I be an interval and let f:I→R be continuous on I. If a,b∈I and if k∈R satisfies f(a)<k<f(b), then there exists a point c∈I btw a and b s.t. f(c)=k.

Front 37

Uniform Continuity Theorem

Back 37

Let I be a closed bounded interval and let f:I∈R be continuous on I. Then f is uniformly continuous on I.

Front 38

Differentiable Implies Continuous Theorem

Back 38

If f:I→R has a derivative at c∈I, then f is continuous at c.