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Let's analyze the continuity of the given function \( f(x) \) at the specified points and intervals.

### 1. \( x = -2 \)
From the graph, at \( x = -2 \), there is a closed dot, meaning the function is defined and there is no break, jump, or vertical asymptote. The graph approaches this point from both sides smoothly, meaning \( f(x) \) is continuous at \( x = -2 \).

### 2. \( x = 3 \)
At \( x = 3 \), we see an open circle, which means that the function is not defined at this point. Thus, the function is **not continuous** at \( x = 3 \).

### 3. Interval \( (-1, 1) \)
In the open interval \( (-1, 1) \), excluding the endpoints, the graph is smooth and unbroken, with no jumps, holes, or asymptotes. Therefore, the function is continuous in this interval.

### 4. Interval \( [-1, 1] \)
In this closed interval, we need to check the endpoints \( x = -1 \) and \( x = 1 \).
- At \( x = -1 \), there is no discontinuity as the function smoothly connects to this point.
- At \( x = 1 \), there is a jump discontinuity due to the gap (open circle at \( x = 1 \)).
Thus, the function is **not continuous** over the entire interval \( [-1, 1] \).

### 5. Interval \( [0, 1) \)
In the half-open interval \( [0, 1) \):
- At \( x = 0 \), the function is continuous.
- The function remains smooth up to but not including \( x = 1 \), where there is a jump discontinuity at the right endpoint.
Thus, the function is continuous over the interval \( [0, 1) \).

### 6. Interval \( (1, +\infty) \)
For \( x > 1 \), the graph is smooth without any breaks, jumps, or vertical asymptotes, meaning that the function is continuous on \( (1, +\infty) \).

### Summary of Continuity:

1. \( x = -2 \): Continuous
2. \( x = 3 \): **Not continuous**
3. \( (-1, 1) \): Continuous
4. \( [-1, 1] \): **Not continuous**
5. \( [0, 1) \): Continuous
6. \( (1, +\infty) \): Continuous

Would you like more detailed explanations for any of these points or intervals?

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Here are 5 related questions to expand on the concept of continuity:
1. What are the conditions for a function to be continuous at a specific point?
2. How does a jump discontinuity affect the continuity of a function?
3. Can a function be continuous over an open interval but not at its endpoints?
4. How do vertical asymptotes relate to discontinuity in a function?
5. What is the difference between removable and non-removable discontinuities?

**Tip:** Continuity at a point requires that the left-hand limit, right-hand limit, and the function value all exist and are equal at that point.

Consider the function f whose graph is given below. Determine whether f is continuous or not on each of the following intervals: 1. x = -2, 2. x = 3, 3. (-1, 1), 4. [-1, 1], 5. [0, 1), 6. (1, +∞).

Explore how to determine the continuity of the function f(x) based on a given graph at specific points and intervals such as x = -2, x = 3, and intervals like (-1, 1) and [0, 1). Learn about key concepts of continuity and the conditions required for a function to be continuous.

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