Define f: R-→ R by if r <1 4 - 2x if a > 1. S(x) = Use the e-d characterization of continuity to show that f is continuous at every xo ER CLcept ro = 1.
Define f: R-→ R by if r <1 4 - 2x if a > 1. S(x) = Use the e-d characterization of continuity to show that f is continuous at every xo ER CLcept ro = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Continuity of a Piecewise Function
**Problem Statement:**
Define \( f: \mathbb{R} \to \mathbb{R} \) by
\[
f(x) = \begin{cases}
x & \text{if } x \leq 1 \\
4 - 2x & \text{if } x > 1.
\end{cases}
\]
Use the \( \epsilon \)-\( \delta \) characterization of continuity to show that \( f \) is continuous at every \( x_0 \in \mathbb{R} \) except \( x_0 = 1 \).
**Explanation:**
This problem involves a piecewise function \( f(x) \) that is defined differently for values of \( x \) less than or equal to 1 and for values greater than 1.
- For \( x \leq 1 \), the function is given by \( f(x) = x \).
- For \( x > 1 \), the function is given by \( f(x) = 4 - 2x \).
To prove continuity at a point using the \( \epsilon \)-\( \delta \) characterization of continuity, you need to show that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x \),
\[ 0 < |x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon. \]
You are asked to demonstrate that this condition holds for all \( x_0 \in \mathbb{R} \) except \( x_0 = 1 \).
**Detailed Steps for Continuity Proof:**
1. **For \( x_0 \leq 1 \) (including \( x_0 < 1 \)):**
- When \( x \leq 1 \), \( f(x) = x \).
- For \( |x - x_0| < \delta \implies |f(x) - f(x_0)| = |x - x_0| \), choose \( \delta = \epsilon \).
- Thus, \( |x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon \), proving continuity for \( x_0 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87a5eb21-df18-43d0-b53f-a372e6adca02%2F99160f49-9c3f-40dc-89ba-ba9d1809d1aa%2F7u81d8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Continuity of a Piecewise Function
**Problem Statement:**
Define \( f: \mathbb{R} \to \mathbb{R} \) by
\[
f(x) = \begin{cases}
x & \text{if } x \leq 1 \\
4 - 2x & \text{if } x > 1.
\end{cases}
\]
Use the \( \epsilon \)-\( \delta \) characterization of continuity to show that \( f \) is continuous at every \( x_0 \in \mathbb{R} \) except \( x_0 = 1 \).
**Explanation:**
This problem involves a piecewise function \( f(x) \) that is defined differently for values of \( x \) less than or equal to 1 and for values greater than 1.
- For \( x \leq 1 \), the function is given by \( f(x) = x \).
- For \( x > 1 \), the function is given by \( f(x) = 4 - 2x \).
To prove continuity at a point using the \( \epsilon \)-\( \delta \) characterization of continuity, you need to show that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x \),
\[ 0 < |x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon. \]
You are asked to demonstrate that this condition holds for all \( x_0 \in \mathbb{R} \) except \( x_0 = 1 \).
**Detailed Steps for Continuity Proof:**
1. **For \( x_0 \leq 1 \) (including \( x_0 < 1 \)):**
- When \( x \leq 1 \), \( f(x) = x \).
- For \( |x - x_0| < \delta \implies |f(x) - f(x_0)| = |x - x_0| \), choose \( \delta = \epsilon \).
- Thus, \( |x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon \), proving continuity for \( x_0 \
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