Is the following piecewise defined function f(x) continuous at x = = 0, [cos(x), x≤0 f(x) = x²+x+1, x>0 ?
Is the following piecewise defined function f(x) continuous at x = = 0, [cos(x), x≤0 f(x) = x²+x+1, x>0 ?
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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![### Continuity of a Piecewise Function
**Question:**
Is the following piecewise defined function \( f(x) \) continuous at \( x = 0 \)?
\[ f(x) = \begin{cases}
\cos(x), & x \leq 0 \\
x^2 + x + 1, & x > 0
\end{cases} \]
Please briefly explain your answer (in 4 sentences or fewer, if possible). Hint: What are the criteria for a function to be continuous at a point \( x_0 \)?
**Explanation:**
For a function to be continuous at a point \( x_0 \), the following three criteria must be satisfied:
1. \( f(x_0) \) is defined.
2. \( \lim_{{x \to x_0}} f(x) \) exists.
3. \( \lim_{{x \to x_0}} f(x) = f(x_0) \).
To determine the continuity at \( x = 0 \):
- Find \( f(0) \) from the piecewise definition, which is \( \cos(0) = 1 \).
- Calculate the left-hand limit: \( \lim_{{x \to 0^-}} \cos(x) = \cos(0) = 1 \).
- Calculate the right-hand limit: \( \lim_{{x \to 0^+}} (x^2 + x + 1) = 1^2 + 0 + 1 = 1 \).
Since all three criteria are satisfied, the function \( f(x) \) is continuous at \( x = 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53539048-d07c-469f-b79c-8ce98d152257%2Fa3b1e0e0-6664-4dc2-a234-6c9b394c41fe%2Fagvxfwu_processed.png&w=3840&q=75)
Transcribed Image Text:### Continuity of a Piecewise Function
**Question:**
Is the following piecewise defined function \( f(x) \) continuous at \( x = 0 \)?
\[ f(x) = \begin{cases}
\cos(x), & x \leq 0 \\
x^2 + x + 1, & x > 0
\end{cases} \]
Please briefly explain your answer (in 4 sentences or fewer, if possible). Hint: What are the criteria for a function to be continuous at a point \( x_0 \)?
**Explanation:**
For a function to be continuous at a point \( x_0 \), the following three criteria must be satisfied:
1. \( f(x_0) \) is defined.
2. \( \lim_{{x \to x_0}} f(x) \) exists.
3. \( \lim_{{x \to x_0}} f(x) = f(x_0) \).
To determine the continuity at \( x = 0 \):
- Find \( f(0) \) from the piecewise definition, which is \( \cos(0) = 1 \).
- Calculate the left-hand limit: \( \lim_{{x \to 0^-}} \cos(x) = \cos(0) = 1 \).
- Calculate the right-hand limit: \( \lim_{{x \to 0^+}} (x^2 + x + 1) = 1^2 + 0 + 1 = 1 \).
Since all three criteria are satisfied, the function \( f(x) \) is continuous at \( x = 0 \).
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