Analysis 2 Notes 3

Question 1

(3D-N10-3.1) Function bounded on a closed interval. Set of all bounded functions denotation.

Answer 1

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Question 2

(3D-N10-3.2) Definition of Partition, Mesh (norm, width), Refinement

Answer 2

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Question 3

(3L-N11-3.1) If P, Q are partitions, and P is a subset of Q, then the mesh of P is greater than or equal to mesh of Q

Answer 3

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Question 4

(3D-N11-3.3) Lower Darboux Sum, Upper Darboux Sum

Answer 4

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Question 5

(3L-N12-3.2) If the partition Q is a refinement of the partition P with one extra point then L(f,P) ≤ L(f,Q), U(f,Q) ≤ U(f,P)

Answer 5

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Question 6

(3L-N12-3.3) If the partition Q is a refinement of the partition P then L(f,P) ≤ L(f,Q), U(f,Q) ≤ U(f,P)

Answer 6

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Question 7

(3T-N12-3.1) For any P, Q ∈ P[a, b], L(f,P) ≤ U(f,Q)

Answer 7

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Question 8

(3D-N13-3.4) Lower, Upper Riemann Integral of f over a closed interval

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Question 9

(3C-N13-3.1) Lower Riemann Intergral less than or equal to Upper Riemann Integral

Answer 9

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Question 10

(3D-N13-3.5) Function Riemann Integrable

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Question 11

(3T-N14-3.2) Riemann’s Criterion for Integrability.

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Question 12

(3T-N15-3.3) Attainment of bounds theorem

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Question 13

(3T-N15-3.4) Continuous functions subset of Riemann Integrable functions

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Question 14

(3D-N15-3.6) Function increasing decreasing

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Question 15

(3D-N16-3.8) Function monotonic

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Question 16

(3T-N16-3.5) If f : [a, b] → R is monotonic on [a, b] then it is Riemann integrable on [a, b].

Answer 16

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Question 17

(3L-N17-3.4) Let the partition Q be a refinement of the partition P with one extra point, and let M be an upper bound for |f| on [a,b]. Then L(f,Q)−2M∥P∥ ≤ L(f,P) ≤ L(f,Q), U(f,Q) ≤ U(f,P) ≤ U(f,Q)+2M∥P∥

Answer 17

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Question 18

(3L-N17-3.5) Let the partition Q be a refinement of the partition P with k extra points, and let M be an upper bound for |f| on [a,b]. Then L(f,Q)−2kM∥P∥ ≤ L(f,P) ≤ L(f,Q), U(f,Q) ≤ U(f,P) ≤ U(f,Q)+2kM∥P∥

Answer 18

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Question 19

(3T-N17-3.6) Let. f∈B[a,b]. Then∀ε>0 ∃δ>0 such that ∥P∥

Answer 19

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Question 20

(3D-N18-3.9) Limiting sequence.

Answer 20

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Question 21

(3T-N18-3.7) Let f ∈ B[ab] and let {Pn} be a limiting sequence of partitions. Then lim n to infinity L(f,Pn) is the LRI. Same with upper sum and Integral (see HW 3)

Answer 21

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Question 22

(3T-N18-3.8) Let f,g ∈ B[a,b] and suppose that f(x) ̸= g(x) only at a finite number of points. Then LRI f(x) = LRI g(x). Same with URI. In particular, f ∈ R[a, b] iff g ∈ R[a, b].

Answer 22

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Question 23

(3T-N19-3.9) Linear Properties. Let f,g ∈ R[a,b]. Then f+g ∈ R[a,b] and (as expected), Then αf ∈ R[a,b] and (as expected). Order Properties.Let f,g ∈ R[a,b] and f ≥ g. Then RI (f) greater than or equal to RI (g). Domain splitting property. Triangle Inequality. Cauchy-Schwarz inequality.

Answer 23

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Question 24

(3O-N20-) What I as an Interval means, plus notation

Answer 24

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Question 25

(3D-N20-3.10) Definition of a function being locally Riemann Integrable.

Answer 25

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Question 26

(3D-N21-3.11) Definition of a function being a primitive

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Question 27

(3T-N21-3.10) Fundamental Theorem of Calculus

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Question 28

(3R-N22-3.1) Fundamental Theorem of Calculus can be proven without __. It suffices to use __

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Question 29

(3D-N22-3.12) Definition of Indefinite Integral

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Question 30

(3T-N23-3.11) Let f ∈ Rloc(I) and let F : I → R be an indefinite integral of f. Then (i) F is continuous on I; (ii) F is differentiable at each interior point x0 ∈ I at which f is continuous; (iii) if f is continuous on I then F is a primitive of f.

Answer 30

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Question 31

(3C-N24-3.2) Let f,g ∈ C[a,b] and let F, G be primitives of f, g respectively. Then …(fill in)

Answer 31

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Question 32

(3C-N25-3.3) Let I be an open interval and let φ : I → R be continuously differen- tiable. Let J be an interval of positive length such that φ(I) ⊂ J and let f : J → R be continuous. Then for any a, b ∈ I

Answer 32

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Question 33

(3T-N25-3.12) Let r be a natural number and let f : [r, +∞) → [0, +∞) be a decreasing function. Then the infinite series (E) f(k) converges iff lim f(x)dx exists.

Answer 33

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