Week 4 and 5

Definition 4.2.3 If f and g are functions defined on A⊆R to R, then their sum, difference and product are defined by:

(f + g)(x) := f(x) + g(x), (f − g)(x) := f(x) − g(x) and (fg)(x) := f(x)g(x), respectively, if g(x) ̸= 0 for all x ∈ A, their quotient is defined by f(x)/g(x):=f/g(x) and the multiple of f by b∈R is defined by (bf)(x) := bf(x), for all x ∈ A.

Sgn(x) =

1 if x>0 0 if x=0 -1 if x<0 F(x) = x/|x|

Sequential Criterion for Limits Theorem 4.1.8Let f :A→R and let c be a cluster point of A. Then TFAE:

1. limx→cf(x)=L 2. For every sequence (xn) in A that con- verges to c such that xn ̸=c for all n∈N, the sequence (f(xn)) converges to L. If we can't to prove that the function does not have a limit at the point, we prove not 1 did not 2

Theorem 4.1.6 Let f :A→R and let c be a cluster point of A. Then TFAE:

1. limx→cf(x)=L. 2. Given any ε-neighbourhood Vε(L) of L, there exists a δ-neighbourhood Vδ(c) of c such that if x ̸= c and x ∈ A∩Vδ(c), then f(x) ∈ Vε(L).

If c is in A but c is not a cluster point or in accA then

C is an iso point and in the set isoA

Sequential characterization of continuous functions requires the limit x of sequence x_nk

To be in the interval on which f is continuous

If xn is Cauchy and has a convergent subsequence xnk converges to x then

Xn converges to x

If (xn) is a convergent sequence of real numbers, then (xn) is a

a Cauchy sequence

Theorem 4.2.9 Let A ⊆ R, f : A → R and let c ∈ R be a cluster point of A. If lim x→c f(x) > 0, then there exists If lim x→c f(x) < 0, then there exists

a neighbourhood Vδ(c) of c such that f(x) > 0 for all x ∈ A ∩ Vδ(c), x ̸= c. a neighbourhood Vδ(c) of c such that f(x) < 0 for all x ∈ A ∩ Vδ(c), x ̸= c.

Remarks 5.1.7 (a) Suppose that f is defined on A but not at c (c 6∈ A) and limx→c f(x) exists, say limx→c f(x) = L. Define F : A ∪ {c} → R by F(x) := L if x = c and =f(x) if x is in A (b) If g : A → R does not have a limit at c, then it is impossible to obtain a function G : A ∪ {c} → R that is continuous at c by defining G(x) := C if x = c and g(x) if x ∈ A.

a) Then F is continuous at c.

Maximum-Minimum Theorem 5.3.4 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. If prove that f(x*) on I where x* is in I is equal to the inf of F(I) and similarly that f(x) on I is equal to sup of F(I) the abs max

Theorem 5.2.2 Let f and g be functions on A ⊆ R to R and let b∈R. Suppose that f and g are continuous on A. Then f +g, f −g, fg and bf

are continuous on A. In addition, if g(x) ̸= 0 for all x ∈ A, then f/g is continuous on A.

A Cauchy sequence of real numbers is

bounded.

Theorem 5.2.1 Let f and g be functions on A ⊆ R to R and let b∈R. Suppose that c∈A and that f and g are continuous at c. Then f +g, f −g, fg and bf are

continuous at c. In addition, if g(x) ̸= 0 for all x ∈ A, then f/g is continuous at c.

Boundedness Theorem 5.3.2 Let I = [a,b] be a closed bounded interval and let f :I →R be continuous on I. Then

f is bounded on I. Note: If any one of the hypotheses (assump- tions, aannames) of the theorem is relaxed, then the theorem fails.

Let A ⊆ R, f : A → R and let c ∈ R be a cluster point of A. If f has a limit at c, then

f is bounded on some neighbourhood of c.

Definition 5.1.5 Let A ⊆ R and let f : A → R. If B is a subset of A, then f is continuous on B if

f is continuous at every point of B.

Sequential Criterion for Continuity Theorem 5.1.3 Let A ⊆ R, f : A → R and c ∈ A. Then f is continuous at c if and only if

for every sequence (xn) in A that converges to c, the sequence (f(xn)) converges to f(c).

Definition 4.1.4 Let A be a subset of R, f :A→R a function and c a cluster point of A. If L∈R, then L is a limit of f at c if

for every ε>0 there exists a δ > 0 such that if x ∈ A and 0 < |x − c| < δ, then |f(x) − L| < ε. 0 < |x-c| is because x can be equal to c Also implies x is in the delta neighborhood of c and f(x) is in the epsilon neighborhood of L

Definition 5.1.1 Let A ⊆ R, f : A → R and c ∈ A. Then f is continuous at c if

given any > 0, there exists δ > 0 such that if x ∈ A and |x − c| < δ, then |f(x) − f(c)| < . If f is not continuous at c, then f is discontinuous at c

Theorem 5.1.2 Let A ⊆ R, f : A → R and c ∈ A. Then f is continuous at c if and only if

given any - neighbourhood V(f(c)) of f(c), there exists a δ-neighbourhood Vδ(c) of c such that if x ∈ A and x ∈ Vδ(c), then f(x) ∈ V(f(c)), i.e. f(A ∩ Vδ(c)) ⊆ V(f(c)).

Definition 3.5.1 A sequence (xn) of real numbers is Cauchy if

if for every e > 0 there exists an N ∈ N such that if n, m ≥ N, then |x_n − x_m| < e

Let A⊆R. A point c∈R is a cluster point (ophopingspunt) of A if

if for every δ > 0 there exists at least one point x ∈ A with x ̸= c such that |x − c| < δ, i.e. if every δ-neighbourhood Vδ(c) = (c − δ, c + δ) of c contains at least one point of A different from c.

Theorem 5.2.7 Let A,B ⊆ R and let f : A → R and g : B → R be functions such that f(A) ⊆ B. If f is continuous on A and g is continuous on B, then the composition g◦f : A → R

is continuous on A.

A sequence of real numbers is convergent if and only if it is

it is a Cauchy sequence.

Theorem 4.2.4 Let f and g be functions on A ⊆ R to R and let c be a cluster point of A. Also, let b∈R. If lim x→c f(x)=L and lim x→c g(x)=M, then

lim (f + g)(x), lim (f − g)(x), lim (f g)(x) lim(bf)(x) exist, and lim(f+g)(x) = L+M, lim(f−g)(x) = L−M, lim(fg)(x) = LM and lim(bf)(x) = bL. In addition, if g(x) ̸= 0 for all x ∈ A and M ̸= 0, then lim(f/g)(x) exists and lim(f/g) (x) = L/M

Recall: Definition 4.1.4 Let A be a subset of R, f : A → R a function and c a cluster point of A. If L ∈ R, then L is a limit of f at c if for every > 0 there exists a δ > 0 such that if x ∈ A and 0 < |x − c| < δ, then |f(x) − L| < e Hence: if c ∈ A is a cluster point of A, then f is continuous at c if and only if

lim x→c f(x) = f(c), and if c ∈ A is not a cluster point of A, then f is automatically continuous at c.

If f : A → R and c is a cluster point of A, then f has at most

one limit at c.

Theorem 5.2.6 Let A, B ⊆ R and let f : A → R and g : B → R be functions such that f(A) ⊆ B. If f is continuous at c ∈ A and g is continuous at b = f(c) ∈ B, then

the composition g◦f : A → R is continuous at c.

Let A ⊆ R, f : A → R and let c ∈ R be a cluster point of A. If a≤f(x)≤b for all x∈A, x /=c, and if lim x→c f(x) exists,

then a≤ lim x→c f(x)≤b

If f has no limit at c,

then f diverges at c.

if f is not bounded on A,

then f is unbounded on A.

Squeeze Theorem (Knyptangstelling) 4.2.7 Let A⊆R, 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 x→c f(x) = L = lim x→c h(x),

then lim x→c g(x) exists and lim g(x) = L. x→c

Let A⊆R, f :A→R such that f(x)≥0 for all x∈A, c∈A and define √f by √f(x):= squarerroot f(x) for x ∈ A. If f is continuous at c (resp., on A), then √f

then √f is continuous at c (resp., on A).

Definition 5.3.1 Let A⊆R and f :A→R. Then f is bounded on A if

there exists a constant M > 0 such that |f(x)|≤M for all x∈A. F is bounded on A if it's range is a bounded set

Sequential characterisation of a cluster point: A number c ∈ R is a cluster point of a set A in R if and only if

there exists a sequence (an) in A such that an ̸=c for all n∈N and lim n→∞ a_n = c.

Discontinuity Criterion 5.1.4 Let A ⊆ R, f : A → R and c ∈ A. Then f is discontinuous at c if and only if

there exists a sequence (xn) in A that converges to c, but the sequence (f(xn)) does not converge to f(c).

Divergence Criteria Theorem 4.1.9 Let f :A→R and let c be a cluster point of A. If L∈R, then f does not have limit L at c if and only if: The function f does not have a limit at c if and only if:

there exists a sequence (xn) in A with xn ̸=c for all n∈N such that (xn) converges to c, but (f(xn)) does not converge to L. there exists a sequence (xn) in A with xn ̸=c for all n∈N such that (xn) converges to c, but (f(xn)) does not converge in R.

Definition 4.2.1 Let A ⊆ R, f : A → R and let c ∈ R be a cluster point of A. Then f is bounded on a neighbourhood of c if

there exists a δ- neighbourhood Vδ(c) of c and a constant M > 0 such that |f(x)| ≤ M for all x ∈ A∩Vδ(c).

Let A⊆R and let f :A→R. Then f has an absolute maximum on A if f has an absolute minimum on A if there

there is a point x∗ ∈A such that f(x∗)≥f(x) for all x∈A, is a point x∗ ∈ A such that f(x∗) ≤ f(x) for all x∈A

Theorem 3.5.8 Every contractive sequence is Cauchy and

therefore convergent

Theorem 5.2.4 Let A⊆R, f :A→R, c∈A and define |f| by |f|(x) := |f(x)| for x ∈ A. If f is continuous at c (resp., on A), then

|f| is continuous at c (resp., on A). (This follows from Exercise 4.2.14.)

A sequence (xn) of real numbers is contractive if there exists a constant C with 0 < C < 1 such that

|x_n+2 − x_n+1| ≤ C|x_n+1 − x_n| for all n ∈ N.