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	1	Determine if the function is continuous at the following points. Justify your conclusions with explanations. Look at the graph
	•	𝑥=0
	•	𝑥=1
	•	𝑥=2


Let's analyze the function's continuity at \( x = 0 \), \( x = 1 \), and \( x = 2 \) based on the graph.

### 1. Continuity at \( x = 0 \):

- **Limit from the left**: As \( x \to 0^- \), the graph approaches \( f(0) = 1 \).
- **Limit from the right**: As \( x \to 0^+ \), the graph also approaches \( f(0) = 1 \).
- **Function value**: The function at \( x = 0 \) is closed (solid circle) and equals 1.

Since the left-hand limit (LHL) equals the right-hand limit (RHL), and both are equal to the function value, the function **is continuous at \( x = 0 \)**.

### 2. Continuity at \( x = 1 \):

- **Limit from the left**: As \( x \to 1^- \), the graph approaches \( f(1) = 2 \).
- **Limit from the right**: As \( x \to 1^+ \), the graph approaches \( f(1) = 0 \).
- **Function value**: There is an open circle at \( x = 1 \), indicating the function is undefined at that point.

Since the left-hand limit does not equal the right-hand limit, and the function is not defined at \( x = 1 \), the function **is not continuous at \( x = 1 \)**.

### 3. Continuity at \( x = 2 \):

- **Limit from the left**: As \( x \to 2^- \), the graph approaches \( f(2) = 0 \).
- **Limit from the right**: As \( x \to 2^+ \), the graph also approaches \( f(2) = 0 \).
- **Function value**: The function at \( x = 2 \) is closed (solid circle) and equals 0.

Since the left-hand limit equals the right-hand limit, and both are equal to the function value, the function **is continuous at \( x = 2 \)**.

### Summary:
- The function **is continuous at \( x = 0 \) and \( x = 2 \)**.
- The function **is not continuous at \( x = 1 \)** due to a jump discontinuity and being undefined.

Would you like more details or have any further questions?

Here are 5 relative questions for further understanding:
1. What conditions must be met for a function to be continuous at a point?
2. What is the difference between a removable discontinuity and a jump discontinuity?
3. How does the function behave near \( x = -2 \), based on the graph?
4. How do limits help in determining the continuity of a function?
5. How would the function change if the point at \( x = 1 \) was closed?

**Tip:** When checking for continuity at a point, always ensure the left-hand limit, right-hand limit, and function value exist and are equal.

Determine if the function is continuous at the following points based on the graph: x = 0, x = 1, x = 2.

Analyze the continuity of a function at x = 0, x = 1, and x = 2 using limits and function behavior from a graph. This problem addresses key concepts of continuity, left-hand and right-hand limits, and jump discontinuities, suitable for students in Grades 9-12.

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