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![Theorem 6.20: Fundamental Theorem of Calculus (Part I) • Statement ○ Let f∈R on [a,b] ○ Define F(x)=∫_a^x▒f(t)dt for x∈[a,b], then § F is continuous on [a,b] ○ Furthermore, if f is continuous at x_0∈[a,b], then § F is differentiable at x_0, and § F^′ (x_0 )=f(x_0 ) • Proof: F is continuous on [a,b] ○ Since f∈R, f is bounded, so ∃M∈R s.t. § |f(t)|≤M,∀a≤t≤b ○ If a≤x y≤b § |F(y)−F(x)|=|∫_y^x▒f(t)dt|≤M(x−y) ○ Given ε 0 § |F(y)−F(x)| ε provided |y−x| ε/M ○ So this shows uniform continuity of F • Proof: F^′ (x_0 )=f(x_0 ) ○ Suppose f is continuous at x_0 ○ Given ε 0,∃δ 0 s.t. § |f(x)−f(x_0 )| ε whenever |x−x_0 | δ for a≤x≤b ○ If x_0−δ s≤x_0≤t x_0+δ where a≤s t≤b § |(F(t)−F(s))/(t−s)−f(x_0 )| § =|(1/(t−s) ∫_s^t▒f(x)dx)−f(x_0 )| § =|(1/(t−s) ∫_s^t▒f(x)dx)−(1/(t−s) ∫_s^t▒f(x_0 )dx)| § =|1/(t−s) ∫_s^t▒(f(x)−f(x_0 ))dx| § |1/(t−s) (t−s)ε|=ε ○ Consequently, F^′ (x_0 )=f(x_0 ) Theorem 6.21: Fundamental Theorem of Calculus (Part II) • Statement ○ Let f∈R on [a,b] ○ If there exists a differentiable function F on [a,b] s.t. F^′=f ○ Then ∫_a^b▒f(x)dx=F(b)−F(a) • Proof ○ Let ε 0 be given ○ Choose a partition P={x_0,x_1,…,x_n } of [a,b] s.t. § U(P,f)−L(P,f) ε ○ Apply the Meal Value Theorem, ∃t_i∈[x_(i−1),x_i ] s.t. § F(x_i )−F(x_(i−1) )=f(t_i )Δx_i where 1≤i≤n ○ Thus,∑_(i=1)^n▒〖f(t_i )Δx_i 〗 forms a telescoping series § ∑_(i=1)^n▒〖f(t_i )Δx_i 〗=F(x_n )−F(x_(n−1) )+F(x_(n−1) )+…−F(x_0 ) § =F(b)+(F(x_(n−1) )−F(x_(n−1) ))+…+(F(x_1 )−F(x_1 ))−F(a) § =F(b)−F(a) ○ Combining the obvious inequalities below § L(P,f)≤∑_(i=1)^n▒〖f(t_i )Δx_i 〗≤U(P,f) § L(P,f)≤∫_a^b▒fdx≤U(P,f) ○ We get § |∑_(i=1)^n▒〖f(t_i )Δx_i 〗−∫_a^b▒fdx| ε § ⇒|F(b)−F(a)−∫_a^b▒fdx| ε ○ Therefore, ∫_a^b▒f(x)dx=F(b)−F(a)](https://shawnzhong.com/wp-content/uploads/2018/05/img_5aee1d791c75e.png)
![Definition 6.1: Riemann Integral • Partition ○ A partition P of a closed interval [a,b] is a finite set of points ○ {x_0,x_1,…,x_n } where a=x_0≤x_1≤…≤x_(n−1)≤x_n=b • Let f be a bounded real function on [a,b], for each partition P of [a,b] ○ Define M_i and m_i to be § M_i=sup┬(x∈[x_(i−1),x_i ] )f(x) § m_i=inf┬(x∈[x_(i−1),x_i ] )f(x) ○ Define the upper sum and lower sum to be § U(P,f)=∑_(i=1)^n▒〖M_i Δx_i 〗 § L(P,f)=∑_(i=1)^n▒〖m_i Δx_i 〗 § where Δx_i=x_i−x_(i−1) ○ Define the upper and lower Reimann integral to be § (∫_a^b▒ ) ̅fdx=inf┬(All P)U(P,f) § ▁(∫_a^b▒ ) fdx=sup┬(All P)L(P,f) • If (∫_a^b▒ ) ̅fdx=▁(∫_a^b▒ ) fdx, then ○ We say that f is Riemann\-integrable on [a,b], and write f∈R ○ Their common value is denoted by∫_a^b▒fdx or ∫_a^b▒f(x)dx • Well-definedness of upper and lower Riemann integral ○ Since f is bounded, ∃m,M∈R s.t. § m≤f(x)≤M (a≤x≤b) ○ Therefore for every partition P of [a,b] § m(b−a)≤L(P,f)≤U(P,f)≤M(b−a) ○ So (∫_a^b▒ ) ̅fdx and ▁(∫_a^b▒ ) fdx are always defined Definition 6.2: Riemann-Stieltjes Integral • Let α be a monotonically increasing function on [a,b] • Let f be a real-valued function bouned on [a,b] • For each partition P of [a,b], define ○ M_i=sup┬(x∈[x_(i−1),x_i ] )f(x) ○ m_i=inf┬(x∈[x_(i−1),x_i ] )f(x) ○ Δα_i=α(x_i )−α(x_(i−1) ) ○ U(P,f,α)=∑_(i=1)^n▒〖M_i Δα_i 〗 ○ L(P,f,α)=∑_(i=1)^n▒〖m_i Δα_i 〗 ○ (∫_a^b▒ ) ̅fdx=inf┬(All P)U(P,f,α) ○ ▁(∫_a^b▒ ) fdx=sup┬(All P)L(P,f,α) • If(∫_a^b▒ ) ̅fdx=▁(∫_a^b▒ ) fdx ○ We denote the common value by∫_a^b▒fdα or ∫_a^b▒f(x)dα(x) ○ This is the Riemann-Stieltjes integral of f with respect to α over [a,b] ○ We say f is integrable with respect to α with on [a,b], and write f∈R(α) • Note ○ When α(x)=x, this is just Riemann integral Definition 6.3: Refinement and Common Refinement • We say that the partition P^∗ is a refinement of P if P^∗⊃P • Given two partitions P_1 and P_2, their common refinement is P_1∪P_2 Theorem 6.4: Properties of Refinement • If P^∗ is a refinement of P, then ○ L(P,f,α)≤L(P^∗,f,α) ○ U(P^∗,f,α)≤U(P,f,α) Theorem 6.5: Properties of Common Refinement • Statement ○ (∫_a^b▒ ) ̅fdx≤▁(∫_a^b▒ ) fdx • Proof Outline ○ Given 2 partitions P_1 and P_2 ○ Let P^∗ be the common refinement ○ Then L(P_1,f,α)≤L(P^∗,f,α)≤U(P^∗,f,α)≤U(P_2,f,α) Theorem 6.6 • Statement ○ f∈R(α) on [a,b] if and only if ○ ∀ε 0, there exists a partition P s.t. U(P,f,α)−L(P,f,α) ε • Proof Outline ○ ∀P,L(P,f,α)≤▁(∫_a^b▒ ) fdx≤(∫_a^b▒ ) ̅fdx≤U(P,f,α) ○ (⟸) If U(P,f,α)−L(P,f,α) ε § Then 0≤(∫_a^b▒ ) ̅fdx−▁(∫_a^b▒ ) fdx ε ○ (⟹) If f∈R(α) § Then ∃P_1,P_2 s.t. □ U(P_1,f,α)−∫_a^b▒fdα ε/2 □ ∫_a^b▒fdα−L(P_1,f,α) ε/2 § Consider their common refinement P § By Theorem 6.4, U(P,f,α)−L(P,f,α) ε Theorem 6.8 • If f is continuous on [a,b], then f∈R(α) on [a,b] Theorem 6.9 • If f is monotonic on [a,b], and α is continuous on [a,b] • Then f∈R(α) on [a,b] Theorem 6.10 • If f is bounded on [a,b] with finitely many points of discontiunity • And α is continuous on these points, then f∈R(α)](https://shawnzhong.com/wp-content/uploads/2018/05/img_5aee155831ac9.png)
![Theorem 5.9: Extended Mean Value Theorem • Statement ○ Given § f and g are continuous real-valued functions on [a,b] § f,g are differentiable on (a,b) ○ Then there is a point x∈(a,b) at which § [f(b)−f(a)] g^′ (x)=[g(b)−g(a)] f^′ (x) • Proof ○ Let h(t)≔[f(b)−f(a)]g(t)−[g(b)−g(a)]f(t), (a≤t≤b) ○ Then h is continuous on [a,b] and differentiable on (a,b) ○ We want to show that h′(x)=0 for some x∈(a,b) ○ By definition of h, h(a)=f(b)g(a)−f(a)g(b)=h(b) ○ If h is constant § h′ (x)=0 on all of (a,b), and we are done ○ If h is not constant § ∃t∈(a,b) s.t. h(t) h(a)=h(b) or h(t) h(a)=h(b) § By Theorem 4.16, ∃x∈(a,b) s.t. § h(x) is either a global maximum or a global minimum § By Theorem 5.8, h′ (x)=0 Theorem 5.10: Mean Value Theorem • Statement ○ If f and g are continuous real-valued functions on [a,b] ○ And f,g are differentiable on (a,b) ○ Then ∃x∈(a,b) s.t. f(b)−f(a)=(b−a) f^′ (x) • Proof ○ Let g(x)=x in Theorem 5.9 Theorem 5.11: Derivative and Monotonicity • Suppose f is differentiable on (a,b) • If f^′ (x)≥0,∀x∈(a,b), then f is monotonically increasing • If f^′ (x)=0,∀x∈(a,b), then f is constant • If f^′ (x)≤0,∀x∈(a,b), then f is monotonically decreasing Theorem 5.15: Taylor s Theorem • Statement ○ Suppose § f is a real-valued function on [a,b] § Fix a positive integer n § f^((n−1) ) is continuous on (a,b) § f^((n) ) (t) exists ∀t∈(a,b) ○ Let α,β∈[a,b], where a≠β ○ Define P(t)=∑_(k=0)^(n−1)▒〖(f^((k) ) (α))/k! (t−α)^k 〗 ○ Then ∃x between α and β s.t. ○ f(β)=P(β)+(f^((n) ) (x))/n! (β−α)^n • Note ○ When n=1, this is the Meal Value Theorem • Proof ○ Without loss of generality, suppose α β ○ Define M∈R by § f(β)=P(β)+M(β−α)^n § Then we want to show that § n!M=f^((n) ) (x) for some x∈[α,β] ○ Define difference function g by § g(t)=f(t)−P(t)−M(t−α)^n, where a≤t≤b § Then g(β)=0 by our choice of M § Taking derivative n times on both side, we get § g^((n) ) (t)=f^((n) ) (t)−n!M, where a≤t≤b § Note that P(t) disappears, since its degree is n−1 ○ Then we only need to show g^((n) ) (x)=0 for some x∈[α,β] § P^((k) ) (α)=f^((k) ) (α),for 0≤k≤n−1, by definition of P § Therefore, g(α)=g^′ (α)=…=g^((n−1) ) (α)=0 § Also, g(β)=0, by definition of M § By the Mean Value Theorem, g^′ (x_1 )=0 for some x_1∈[α,β] § g^′ (α)=0, so g^′′ (x_2 )=0 for some x_2∈[α,x_1 ] § After n steps, g^((n) ) (x_n )=0 for some x_n∈[α,x_(n−1) ] § So, x_n∈[α,β]](https://shawnzhong.com/wp-content/uploads/2018/05/img_5aebe3879b0e7.png)
![Definition 5.1: Derivative • Let f be defined (and real-valued) on [a,b] • ∀x∈[a,b], let ϕ(t)=(f(t)−f(x))/(t−x) (a t b, t≠x) • Define f^′ (x)=(lim)_(t→x)ϕ(t), provided that this limit exists • f′ is called the derivative of f • If f′ is defined at point x, f is differentiable at x • If f′ is defined ∀x∈E⊂[a,b], then f is differentiable on E Theorem 5.2: Differentiability Implies Continuity • Statement ○ Let f be defined on [a,b] ○ If f is differentiable at x∈[a,b] then f is continuous at x • Proof ○ lim_(t→x)(f(t)−f(x))=lim_(t→x)((f(t)−f(x))/(t−x) (t−x))=lim_(t→x)(f^′ (x)(t−x))=0 ○ So lim_(t→x)f(t)=f(x) Theorem 5.5: Chain Rule • Statement ○ Given § f is continuous on [a,b], and f^′ (x) exists at x∈[a,b] § g is defined on I⊃im(f), and g is differentiable at f(x) ○ If h(t)=g(f(t)) (a≤t≤b), then ○ h is differentiable at x, and h^′ (x)=g^′ (f(x))⋅f^′ (x) • Proof ○ Let y=f(x) ○ By the definition of derivative § f(t)−f(x)=(t−x)(f^′ (x)+u(t)), where t∈[a,b], lim_(t→x)u(t)=0 § g(s)−g(y)=(s−y)(g^′ (y)+v(s)), where s∈I, lim_(s→y)v(s)=0 ○ Let s=f(t), then § h(t)−h(x)=g(f(t))−g(f(x)) § =(f(t)−f(x))(g^′ (y)+v(s)) § =(t−x)(f^′ (x)+u(t))(g^′ (y)+v(s)) ○ If t≠x, then § (ht)−hx))/(t−x)=(f^′ (x)+u(t))(g^′ (y)+v(s)) ○ As t→x § u(t)→0, and v(s)→s § So s=f(t)→f(x)=y by continuity § Therefore h′ (x)=lim_(t→x)〖(ht)−hx))/(t−x)〗=f^′ (x) g^′ (y)=g^′ (f(x)) f^′ (x) Definition 5.7: Local Maximum and Local Minimum • Let X be a metric space, f:X→R • f has a local maximum at p∈X if ∃δ 0 s.t. ○ f(q)≤f(p),∀q∈X s.t. d(p,q) δ • f has a local minimum at p∈X if ∃δ 0 s.t. ○ f(q)≥f(p),∀q∈X s.t. d(p,q) δ Theorem 5.8: Local Extrema and Derivative • Statement ○ Let f be defined on [a,b] ○ If f has a local maximum (or minimum) at x∈(a,b) ○ Then f^′ (x)=0 if it exists • Proof ○ By Definition 5.7, choose δ, then § a x−δ x x+δ b ○ Suppose x−δ t x § (f(t)−f(x))/(t−x)≥0 § Let t→x (with t x), then f^′ (x)≥0 ○ Suppose x t x+δ § (f(t)−f(x))/(t−x)≤0 § Let t→x (with t x), then f^′ (x)≤0 ○ Therefore f^′ (x)=0](https://shawnzhong.com/wp-content/uploads/2018/05/img_5aea9e3142970.png)
![Definition 2.45: Connected Set • Let X be a metric space, and A,B⊂X • A and B are separated if ○ A∪B ̅=∅ and A ̅∪B=∅ ○ i.e. No point of A lies in the closure of B and vice versa • E⊂X is connected if ○ E is not a union of two nonempty separated sets Theorem 2.47: Connected Subset of R • Statement ○ E⊂R is connected if and only if E has the following property ○ If x,y∈E and x z y, then z∈E • Proof (⟹) ○ By way of contrapositive, suppose ∃x,y∈E, and z∈(x,y) s.t. z∉E ○ Let A_z=E∩(−∞,z) and B_z=E∩(z,+∞) ○ Then A_z and B_z are separated and E=A_z∪B_z ○ Therefore E is not connected • Proof (⟸) ○ By way of contrapositive, suppose E is not connected ○ Then there are nonempty separated sets A and B s.t. E=A∪B ○ Let x∈A,y∈B. Without loss of generality, assume x y ○ Let z≔sup(A∩[x,y]). Then by Theorem 2.28, z∈A ̅ ○ By definition of E, z∉B. So, x≤z y ○ If z∉A § x∈A and z∉A § ⇒x z y § ⇒z∉E ○ If z∈A § Since A and B are separated, z∉B ̅ § So ∃z_1 s.t. z z_1 y and z_1∉B § Then x z_1 y, so z_1∉E Theorem 4.22: Continuous Mapping of Connected Set • Statement ○ Let X,Y be metric spaces ○ Let f:X→Y be a continuous mapping ○ If E⊂X is connected then f(E)⊂Y is also connected • Proof ○ Suppose, by way of contradiction, that f(E) is not connected ○ "i.e." f(E)=A∪B, where A,B⊂Y are nonempty and separated ○ Let G≔E∩f^(−1) (A) and H≔E∩f^(−1) (B) ○ Then E=G∪H, where G,H≠∅ ○ Since A⊂A ̅, we have G⊂f^(−1) (A ̅ ) ○ Since f is continuous and A ̅ is closed, f^(−1) (A ̅ ) is also closed ○ Therefore G ̅⊂f^(−1) (A ̅ ), and hence f(G ̅ )⊂A ̅ ○ Since f(H)=B and A ̅∩B=∅, we have G ̅∩H=∅ ○ Similarly, G∩H ̅=∅ ○ So, G and H are separated ○ This is a contradiction, therefore f(E) is connected Theorem 4.23: Intermediate Value Theorem • Statement ○ Let f:R→R be continuous on [a,b] ○ If f(a) f(b) and if c statifies f(a) c f(b) ○ Then ∃x∈(a,b) s.t. f(x)=c • Proof ○ By Theorem 2.47, [a,b] is connected ○ By Theorem 4.22, f([a,b]) is a connected subset of R ○ By Theorem 2.47, the result follows](https://shawnzhong.com/wp-content/uploads/2018/05/img_5aea9aa22812b.png)