### All Introduction to Analysis Resources

## Example Questions

### Example Question #1 : Integrability Of Real Numbers (R)

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

**Possible Answers:**

, , , and be bounded

, , , and be bounded

, , , and be bounded

, , , and

, , , and

**Correct answer:**

, , , and be bounded

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 represents the upper integral and the following are defined:

1. The upper integral of on is

where is a partition of .

2. The lower integral of on is

where is a partition of .

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

if and only if , , , and be bounded.

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if , , , and be bounded.

### Example Question #2 : Integrability Of Real Numbers (R)

What term has the following definition.

, and . Over the interval is a set of points such that

**Possible Answers:**

Refinement of a partition

Lower Riemann sum

Upper Riemann sum

Partition

Norm

**Correct answer:**

Partition

By definition

If , and .

A *partition* over the interval is a set of points such that

.

Therefore, the term that describes this statement is partition.

### Example Question #1 : Riemann Integral, Riemann Sums, & Improper Riemann Integration

What term has the following definition.

The __________ of a partition is

**Possible Answers:**

Lower Riemann sum

Upper Riemann Sum

Refinement of a partition

Partition

Norm

**Correct answer:**

Norm

By definition

If , and .

A partition over the interval is a set of points such that

.

Furthermore,

The *norm *of the partition

is

Therefore, the term that describes this statement is norm.

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