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?

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.