https://file.aiquickdraw.com/mathbot/file/17254834503425tsj3pc6.jpg

The piecewise function provided in the image is:

\[
f(x) = 
\begin{cases} 
2x + 1, & \text{if } x \leq -1 \\
3x, & \text{if } -1 < x < 1 \text{ and at } x = -1 \text{ and } x = 1 \\
2x - 1, & \text{if } x \geq 1 
\end{cases}
\]

Let's analyze this function step-by-step:

### 1. Continuity Check
To check the continuity at the points where the definition of the function changes, specifically at \(x = -1\) and \(x = 1\), we need to ensure that the left-hand limit (LHL), right-hand limit (RHL), and the function value at those points are equal.

#### At \(x = -1\):
- Left-hand limit: \(\lim\limits_{x \to -1^-} f(x) = 2(-1) + 1 = -2 + 1 = -1\)
- Right-hand limit: \(\lim\limits_{x \to -1^+} f(x) = 3(-1) = -3\)
- \(f(-1)\) according to the second case is \(3(-1) = -3\)

Since LHL \(\neq\) RHL, the function is **not continuous** at \(x = -1\).

#### At \(x = 1\):
- Left-hand limit: \(\lim\limits_{x \to 1^-} f(x) = 3(1) = 3\)
- Right-hand limit: \(\lim\limits_{x \to 1^+} f(x) = 2(1) - 1 = 2 - 1 = 1\)
- \(f(1)\) according to the second case is \(3(1) = 3\)

Since LHL \(\neq\) RHL, the function is **not continuous** at \(x = 1\).

### 2. Differentiability Check
For a function to be differentiable at a point, it must be continuous there, and the left-hand derivative must equal the right-hand derivative. Since we already found that the function is not continuous at \(x = -1\) and \(x = 1\), the function is also **not differentiable** at these points.

Would you like more details on any part of the solution, or do you have any specific questions?

### 5 Relative Questions:
1. How would you determine if a piecewise function is differentiable at a given point?
2. Can you give an example of a piecewise function that is both continuous and differentiable at all points?
3. What is the significance of continuity in real-world applications of piecewise functions?
4. How can you graph a piecewise function to visualize its continuity and differentiability?
5. What are the general conditions for a function to be continuous and differentiable?

### Tip:
When dealing with piecewise functions, always carefully analyze the points where the definition changes to check for continuity and differentiability.

Analyze the continuity and differentiability of the piecewise function provided in the image.

Explore the analysis of the continuity and differentiability of a piecewise function with detailed explanations. Learn about key concepts like left-hand and right-hand limits, and how they apply to determining continuity and differentiability.

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