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.