Question

Math

Posted 5 months ago

`Let K be a nonempty subset of R^n, where n > 1. Which of the following statements must be true? I. If K is compact, then every continuous real-valued function defined on K is bounded. II. If every continuous real-valued function defined on K is bounded, then K is compact. III. If K is compact, then K is connected.`

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

To determine the truth of the statements, we must use the definitions of compactness, boundedness, and connectedness in $\mathbb{R}^n$

step 2

Statement I: If $K$ is compact, then every continuous real-valued function defined on $K$ is bounded. This is true by the Extreme Value Theorem

step 3

Statement II: If every continuous real-valued function defined on $K$ is bounded, then $K$ is compact. This is not necessarily true; boundedness alone does not imply compactness

step 4

Statement III: If $K$ is compact, then $K$ is connected. This is not necessarily true; compactness does not imply connectedness

Answer

Statement I is true; Statements II and III are false.

Key Concept

Compactness and Continuous Functions

Explanation

A compact subset of $\mathbb{R}^n$ ensures that every continuous function defined on it is bounded (Extreme Value Theorem), but compactness does not imply connectedness, and boundedness of functions alone does not imply compactness of the set.

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