Flashcards in Analysis III Deck (17):

1

## Step function

### 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.

2

## Refinement of a partition P

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

3

## Integral of a step function

###
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).

4

## FTC - step fns

###
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).

5

## Sup norm

###
Let 𝑓 ∈ B[a,b]. The sup norm of 𝑓, ||⋅||_∞: B[a,b] → ℝ, is s.t.

||𝑓||_∞ = sup_{𝑥 ∈ [a,b]} |𝑓(𝑥)|

6

## Regulated function

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

7

## Converge uniformly

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

8

## Uniformly continuous

### 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.

9

## Integral of a regulated function

###
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 → ∞.

10

## Weakly increasing

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

11

## Indefinite integral of regulated fn

###
Let 𝑓 ∈ R[a,b]. The indefinite integral of 𝑓 is F: [a,b] → ℝ,

F: 𝑥 ↦ ∫_{𝑥,a} 𝑓.

12

## FTC1

### 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).

13

## FTC2

###
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).

14

## Riemann's sum

###
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.

15

## Bonzano-Weierstrass Theorem

### Every bounded sequence in ℝⁿ has a convergent subsequence.

16

## Continuous

### 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.

17