7. Let FCR be a nonempty closed set and define g(x) = inf{a- al : a € F}. %3D Show that g is continuous on all of R and that g(x) 0 for any r 4 F.
7. Let FCR be a nonempty closed set and define g(x) = inf{a- al : a € F}. %3D Show that g is continuous on all of R and that g(x) 0 for any r 4 F.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How would you prove #7?
![### Continuous Functions and Metric Spaces
**Problem 7:**
Let \( F \subseteq \mathbb{R} \) be a nonempty closed set and define
\[
g(x) = \inf\{ |x - a| : a \in F \}.
\]
Show that \( g \) is continuous on all of \(\mathbb{R}\) and that \( g(x) \neq 0 \) for any \( x \notin F \).
### Explanation:
In this exercise, we are working with the concept of infimum (greatest lower bound) to define the function \( g(x) \). The function \( g \) measures the shortest distance from any point \( x \in \mathbb{R} \) to the set \( F \).
#### Key Points:
- **Closed Set**: \( F \) being closed in \(\mathbb{R}\) ensures that it contains all its limit points.
- **Continuity of \( g \)**: You are required to demonstrate that \( g \) does not have any disjoint points on \(\mathbb{R}\), which means it should smoothly transition without jumps or breaks.
- **Non-zero condition**: For any \( x \notin F \), \( g(x) > 0 \) because if \( x \) is not in \( F \), there is a positive distance to the nearest point in \( F \).
### Approach:
1. **Prove Continuity**: Use the definition of continuity in terms of \(\epsilon\)-\(\delta\) and properties of closed sets to argue that small changes in \( x \) result in small changes in \( g(x) \).
2. **Ensure \( g(x) \neq 0 \)**: Argue that for \( x \notin F \), the distance function \( |x - a| \) must be strictly positive due to the definition of a closed set and the properties of real numbers.
By working through these points, the solution will affirm the properties of function \( g \) based on the principles of metric space and real analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83dd687e-6aa5-4cab-9f16-2444a90701da%2F2faa74e3-704f-43c9-baa8-6a4172ae1251%2Fcbreh5i.jpeg&w=3840&q=75)
Transcribed Image Text:### Continuous Functions and Metric Spaces
**Problem 7:**
Let \( F \subseteq \mathbb{R} \) be a nonempty closed set and define
\[
g(x) = \inf\{ |x - a| : a \in F \}.
\]
Show that \( g \) is continuous on all of \(\mathbb{R}\) and that \( g(x) \neq 0 \) for any \( x \notin F \).
### Explanation:
In this exercise, we are working with the concept of infimum (greatest lower bound) to define the function \( g(x) \). The function \( g \) measures the shortest distance from any point \( x \in \mathbb{R} \) to the set \( F \).
#### Key Points:
- **Closed Set**: \( F \) being closed in \(\mathbb{R}\) ensures that it contains all its limit points.
- **Continuity of \( g \)**: You are required to demonstrate that \( g \) does not have any disjoint points on \(\mathbb{R}\), which means it should smoothly transition without jumps or breaks.
- **Non-zero condition**: For any \( x \notin F \), \( g(x) > 0 \) because if \( x \) is not in \( F \), there is a positive distance to the nearest point in \( F \).
### Approach:
1. **Prove Continuity**: Use the definition of continuity in terms of \(\epsilon\)-\(\delta\) and properties of closed sets to argue that small changes in \( x \) result in small changes in \( g(x) \).
2. **Ensure \( g(x) \neq 0 \)**: Argue that for \( x \notin F \), the distance function \( |x - a| \) must be strictly positive due to the definition of a closed set and the properties of real numbers.
By working through these points, the solution will affirm the properties of function \( g \) based on the principles of metric space and real analysis.
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