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Flashcards: Integrability of Real Numbers (R)

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

, $b\ \epsilon\ \mathbb{R}$ , a>b , and $f:[a,b]\rightarrow \mathbb{R}$

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{Z}$ be bounded

, $b\ \epsilon\ \mathbb{Z}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded

, $b\ \epsilon\ \mathbb{R}$ , a>b , and $f:[a,b]\rightarrow \mathbb{Z}$

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Flashcards: Integrability of Real Numbers (R)

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

Your answer:

, $b\ \epsilon\ \mathbb{R}$ , a>b , and $f:[a,b]\rightarrow \mathbb{R}$

Your answer:

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{Z}$ be bounded

Your answer:

, $b\ \epsilon\ \mathbb{Z}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded

Your answer:

, $b\ \epsilon\ \mathbb{R}$ , a>b , and $f:[a,b]\rightarrow \mathbb{Z}$

Correct answer:

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded

Explanation:

Using the definition for Riemann sums to define the upper and lower integrals of a function answers the question.

According the the Riemann sum where U(f,P) represents the upper integral and L(f,P) the following are defined:

1. The upper integral of on [a,b] is

$(U)\int_a^bf(x)dx:=\text{inf}\begin{Bmatrix} U(f,p)) \end{Bmatrix}$ where is a partition of [a,b] .

2. The lower integral of on [a,b] is

$(L)\int_a^bf(x)dx:=\text{sup}\begin{Bmatrix} L(f,p)) \end{Bmatrix}$ where is a partition of [a,b] .

3. If 1 and 2 are the same then the integral is said to be

$\int_a^bf(x):=(U)\int_a^bf(x)dx=(L)\int_a^bf(x)dx$

if and only if , $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if , $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

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All Introduction to Analysis Resources

Flashcards: Integrability of Real Numbers (R)

Flashcards: Integrability of Real Numbers (R)

Flashcards: Integrability of Real Numbers (R)

Flashcards: Integrability of Real Numbers (R)

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Flashcards: Integrability of Real Numbers (R)

Flashcards: Integrability of Real Numbers (R)

Flashcards: Integrability of Real Numbers (R)

Flashcards: Integrability of Real Numbers (R)

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