Analysis III

Question 1

Step function

Answer 1

φ: [a,b] → ℝ is a step function if there exists a finite set P ⊂ [a,b] s.t. P = {a = 𝑝₀, 𝑝₁, 𝑝₂, … , 𝑝ₖ₋₁, 𝑝ₖ = b}, 𝑝₀ < 𝑝₁ < 𝑝₂ < … < 𝑝ₖ₋₁ < 𝑝ₖ, and φ|_(𝑝ᵢ₋₁, 𝑝ᵢ) = 𝑐ᵢ for 𝑖 = 1, 2, …, k-1, k. Then P is a partition compatible with ϕ.

Question 2

Refinement of a partition P

Answer 2

If Q and P are two partitions of [a,b] and Q ⊃ P, then Q is a refinement of P.

Question 3

Integral of a step function

Answer 3

For φ ∈ S[a,b], P = {𝑝ᵢ} ᵢ₌₀ ᵏ a partition compatible with φ s.t. φ|(𝑝ᵢ₋₁, 𝑝ᵢ) =𝑐ᵢ. Then ∫{a,b}: S[a,b] → ℝ is
∫_{a,b} φ = ∑ᵢ₌₁ ᵏ 𝑐ᵢ(𝑝ᵢ - 𝑝ᵢ₋₁).

Question 4

FTC - step fns

Answer 4

Let φ ∈ S[a,b], P partition of [a,b] compatible with φ. Consider I: [a,b] → ℝ. I(x) = ∫_{a,x} φ. Then

1) I is cts on [a,b]
2) I is diffble on ∪{𝑖=1, k} (𝑝ᵢ₋₁, 𝑝ᵢ), and, ∀𝑥 ∈ ∪{𝑖=1, k} (𝑝ᵢ₋₁, 𝑝ᵢ), I’(𝑥) = φ(𝑥).

Question 5

Sup norm

Answer 5

Let 𝑓 ∈ B[a,b]. The sup norm of 𝑓, ||⋅||∞: B[a,b] → ℝ, is s.t.
||𝑓||∞ = sup_{𝑥 ∈ [a,b]} |𝑓(𝑥)|

Question 6

Regulated function

Answer 6

A function 𝑓 on [a,b] is regulated if, ∀ ε>0, ∃ φ ∈ S[a,b] s.t. ||φ - 𝑓||_∞ < ε.

Question 7

Converge uniformly

Answer 7

A sequence (φₙ){n ≥ 1} ⊂ S[a,b] converges uniformly to a function 𝑓: [a,b] → ℝ if lim{n → ∞} ||𝑓 - φₙ||_∞ = 0.

Question 8

Uniformly continuous

Answer 8

A function 𝑓: A ⊂ ℝ → ℝ is uniformly cts on A, if ∀ ε>0, ∃ δ s.t. ∀ 𝑥,𝑦 ∈ A, |𝑥 - 𝑦| < δ and |𝑓(𝑥) - 𝑓(𝑦)| < ε.

Question 9

Integral of a regulated function

Answer 9

Let 𝑓 ∈ R[a,b]. Then we define ∫_{a,b}: R[a,b] → ℝ as

∫{a,b} 𝑓 = lim{n → ∞} ∫{a,b} φₙ, where (φₙ){n ≥ 1} ⊂ S[a,b] converges to 𝑓 uniformly as n → ∞.

Question 10

Weakly increasing

Answer 10

A fn 𝑓: [a,b] → ℝ is weakly increasing if 𝑦 ≥ 𝑥 ⇒ 𝑓(𝑦) ≥ 𝑓(𝑥).

Question 11

Indefinite integral of regulated fn

Answer 11

Let 𝑓 ∈ R[a,b]. The indefinite integral of 𝑓 is F: [a,b] → ℝ,
F: 𝑥 ↦ ∫_{𝑥,a} 𝑓.

Question 12

FTC1

Answer 12

Let 𝑓 ∈ R[a,b] s.t. 𝑓 is cts at 𝑐 ∈ (a,b). Then F(𝑥) = ∫ _{a,b} 𝑓 is diffble at 𝑐 and F’(𝑐) = 𝑓(𝑐).

Question 13

FTC2

Answer 13

Let 𝑓: [a,b] → ℝ be cts, and let 𝑔: [a,b] → ℝ be diffble, with 𝑔’(𝑥) = 𝑓(𝑥) ∀ x ∈ [a,b] (right deriv at a, left deriv at b). Then
∫_{a,b}𝑓 = 𝑔(b) - 𝑔(a).

Question 14

Riemann’s sum

Answer 14

For 𝑓 ∈ R[a,b], Riemann’s sum is

R_{𝑓} ⁽ⁿ⁾ = ∑_{j=1,n} (b-a)f(aⱼ)/n, where aⱼ = a + (b-a)*j/n.

Question 15

Bonzano-Weierstrass Theorem

Answer 15

Every bounded sequence in ℝⁿ has a convergent subsequence.

Question 16

Continuous

Answer 16

𝑓: E → ℝ is contiunous at 𝑐 ∈ E if ∀ ε>0, ∃ δ>0 s.t. (𝑥 ∈ E and |𝑥 - c| < δ) ⇒ |𝑓(𝑥) - 𝑓(𝑐)| < ε.

Question 17

Sequentially continuous

Answer 17

𝑓 : E → ℝ is sequentially
continuous at 𝑐 ∈ E if whenever (xₙ) ∈ E is such that xₙ → 𝑐 as
n → ∞, then 𝑓(xₙ) → 𝑓(c) as n → ∞.