Please show supporting work to verify the three parts of the definition of Continuity **Function Continuity Analysis**The problem presented is to determine whether the function \( f(x) \) is continuous at \( x = 5 \). The function is defined as follows:\[f(x) = \begin{cases} \frac{1}{5}x + 4, & \text{for } x \leq 5 \\2x - 1, & \text{for } x > 5 \end{cases}\]### Options for Continuity Evaluation:- **A.** The given function is not continuous at \( x = 5 \) because \(\lim_{x \to 5} f(x)\) does not exist.- **B.** The given function is continuous at \( x = 5 \) because the limit is 5.- **C.** The given function is not continuous at \( x = 5 \) because \( f(5) \) does not exist.- **D.** The given function is continuous at \( x = 5 \) because \(\lim_{x \to 5} f(x)\) does not exist.**Note**: The problem appears as a multiple-choice question for students to evaluate the continuity of the given piecewise function at a specific point. There are no accompanying graphs or diagrams. **Continuity of Functions**A function \( f \) is continuous at \( x = a \) if:1. \( f(a) \) exists *(A point appears on the graph when \( x = a \).)*2. \( \lim_{x \to a} f(x) \) exists *(The curve has the same height when approaching \( x = a \) from either side.)*3. \( \lim_{x \to a} f(x) = f(a) \) *(The point occupies the same position noted in #2.)*A function \( f \) is continuous over an interval \( I \) if it is continuous at every point in \( I \).A function \( f \) that fails one or more of these conditions is said to be discontinuous at \( x = a \).**Basic idea:** A function is continuous over an interval if the graph has no breaks, skips, or gaps.

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1 Is the function given by f(x) = 5x+4, for x ≤5, 2x-1, for x>5, Choose the correct answer below. continuous at x = 5? Why or why not? A. The given function is not continuous at x = 5 because lim f(x) does not exist. X→5 OB. The given function is continuous at x = 5 because the limit is 5. OC. The given function is not continuous at x = 5 because f(5) does not exist. O D. The given function is continuous at x = 5 because lim f(x) does not exist. X-5

1 Is the function given by f(x) = 5x+4, for x ≤5, 2x-1, for x>5, Choose the correct answer below. continuous at x = 5? Why or why not? A. The given function is not continuous at x = 5 because lim f(x) does not exist. X→5 OB. The given function is continuous at x = 5 because the limit is 5. OC. The given function is not continuous at x = 5 because f(5) does not exist. O D. The given function is continuous at x = 5 because lim f(x) does not exist. X-5

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Please show supporting work to verify the three parts of the definition of Continuity

**Function Continuity Analysis**

The problem presented is to determine whether the function \( f(x) \) is continuous at \( x = 5 \). The function is defined as follows:

\[
f(x) =
\begin{cases}
\frac{1}{5}x + 4, & \text{for } x \leq 5 \\
2x - 1, & \text{for } x > 5
\end{cases}
\]

### Options for Continuity Evaluation:

- **A.** The given function is not continuous at \( x = 5 \) because \(\lim_{x \to 5} f(x)\) does not exist.
- **B.** The given function is continuous at \( x = 5 \) because the limit is 5.
- **C.** The given function is not continuous at \( x = 5 \) because \( f(5) \) does not exist.
- **D.** The given function is continuous at \( x = 5 \) because \(\lim_{x \to 5} f(x)\) does not exist.

**Note**: The problem appears as a multiple-choice question for students to evaluate the continuity of the given piecewise function at a specific point. There are no accompanying graphs or diagrams.

**Continuity of Functions**

A function \( f \) is continuous at \( x = a \) if:

1. \( f(a) \) exists
*(A point appears on the graph when \( x = a \).)*

2. \( \lim_{x \to a} f(x) \) exists
*(The curve has the same height when approaching \( x = a \) from either side.)*

3. \( \lim_{x \to a} f(x) = f(a) \)
*(The point occupies the same position noted in #2.)*

A function \( f \) is continuous over an interval \( I \) if it is continuous at every point in \( I \).

A function \( f \) that fails one or more of these conditions is said to be discontinuous at \( x = a \).

**Basic idea:** A function is continuous over an interval if the graph has no breaks, skips, or gaps.

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