Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Looking at photo1, can you prove the circled function {f.g} - Continue and explain with the starting HINT on photo2 in detail
Transcribed Image Text:### Combinations of Continuous Functions
For a subset \(D \subseteq \mathbb{R}^p\),
**Theorem:**
Let \( f, g: D \rightarrow \mathbb{R}^q \) be continuous at \( a \in \mathbb{R}^p \).
Then:
1. \( f + g, f - g, cf, fg \) are continuous at \( a \).
Additionally, \( \frac{f}{g} \) is continuous at \( a \) provided \( g(a) \neq 0 \).
2. If \( f \) is continuous at \( a \) and \( g \) is continuous at \( b = f(a) \), then \( g \circ f \) is continuous at \( a \).
![### Continuity of Product of Continuous Functions
This blackboard contains a proof related to the continuity of the product of two functions. Here is a transcription and explanation of the content:
1. **Definitions and Assumptions:**
- Functions \( f, g \) with domain \( D \subset \mathbb{R}^p \) are continuous, where \( a \in \mathbb{R}^p \).
- The focus is on proving that \( f \cdot g \) is continuous at \( a \).
2. **Proof Structure:**
- **Continuity Condition:** For \( f \) and \( g \) to be continuous at \( a \):
- Given \( \varepsilon > 0 \), there exists \( \delta_1 > 0 \) such that for all \( x \in \mathbb{R} \), if \( \| x - a \| < \delta_1 \), then \( \| f(x) - f(a) \| < \varepsilon \).
- Similarly, there is \( \delta_2 > 0 \) such that \( \| g(x) - g(a) \| < \varepsilon \).
- **Combined Condition:** Choose \( \delta = \min(\delta_1, \delta_2) \). Thus, if \( \| x - a \| < \delta \), both continuity conditions are satisfied.
3. **Mathematical Expression:**
- Shown centrally is a bound expression for the product of \( f \) and \( g \):
\[
\| f(x) \cdot g(x) - f(a) \cdot g(a) \|
\]
- This expression expands using properties of modulus and distribution:
\[
= \| (f(x) - f(a)) \cdot g(a) + f(x) \cdot (g(x) - g(a)) \|
\]
- Further breaking down:
\[
= \| (f(x) - f(a)) \cdot g(a) \| + \| f(x) \cdot (g(x) - g(a)) \|
\]
This proof demonstrates that if \( f \) and \( g \) are continuous at a point \( a \), then their product \( f \cdot g \](https://content.bartleby.com/qna-images/question/f5203d42-0da6-4b1b-a1a9-5b37798d2cd6/5842ec03-ae9a-4e4c-94f4-e02856131802/rm8va7_processed.jpeg)
Transcribed Image Text:### Continuity of Product of Continuous Functions
This blackboard contains a proof related to the continuity of the product of two functions. Here is a transcription and explanation of the content:
1. **Definitions and Assumptions:**
- Functions \( f, g \) with domain \( D \subset \mathbb{R}^p \) are continuous, where \( a \in \mathbb{R}^p \).
- The focus is on proving that \( f \cdot g \) is continuous at \( a \).
2. **Proof Structure:**
- **Continuity Condition:** For \( f \) and \( g \) to be continuous at \( a \):
- Given \( \varepsilon > 0 \), there exists \( \delta_1 > 0 \) such that for all \( x \in \mathbb{R} \), if \( \| x - a \| < \delta_1 \), then \( \| f(x) - f(a) \| < \varepsilon \).
- Similarly, there is \( \delta_2 > 0 \) such that \( \| g(x) - g(a) \| < \varepsilon \).
- **Combined Condition:** Choose \( \delta = \min(\delta_1, \delta_2) \). Thus, if \( \| x - a \| < \delta \), both continuity conditions are satisfied.
3. **Mathematical Expression:**
- Shown centrally is a bound expression for the product of \( f \) and \( g \):
\[
\| f(x) \cdot g(x) - f(a) \cdot g(a) \|
\]
- This expression expands using properties of modulus and distribution:
\[
= \| (f(x) - f(a)) \cdot g(a) + f(x) \cdot (g(x) - g(a)) \|
\]
- Further breaking down:
\[
= \| (f(x) - f(a)) \cdot g(a) \| + \| f(x) \cdot (g(x) - g(a)) \|
\]
This proof demonstrates that if \( f \) and \( g \) are continuous at a point \( a \), then their product \( f \cdot g \
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