From the graph of the function $f(x)$, we need to find and classify all points of discontinuity.

Steps to analyze the discontinuities:

1. Identify points of discontinuity visually: Discontinuities occur where the graph has breaks, jumps, or holes.
2. Classify each discontinuity: Discontinuities are usually classified as:
   • Removable discontinuity: The graph has a hole, and the limit exists, but the function value is not defined at that point or differs from the limit.
   • Jump discontinuity: The graph jumps at a point, i.e., the left and right-hand limits exist but are not equal.
   • Infinite discontinuity: The graph approaches infinity at the discontinuity, indicating a vertical asymptote.

Observations:

• At $x = -2$: There is a removable discontinuity. There is a hole in the graph at $x = -2$, but the function value is defined just below the hole.
• At $x = 1$: There is a jump discontinuity. The graph has two different values when approaching from the left and the right, indicating a jump.
• At $x = 2$: There is an infinite discontinuity. The graph shoots off towards infinity, indicating a vertical asymptote.

Summary of discontinuities:

1. At $x = -2$: Removable discontinuity (hole in the graph).
2. At $x = 1$: Jump discontinuity (function has a jump between two different values).
3. At $x = 2$: Infinite discontinuity (vertical asymptote).

Would you like more details on any of these classifications or explanations?

5 Related Questions:

1. What is the definition of a removable discontinuity, and how is it different from a jump discontinuity?
2. Can you explain how limits behave around a jump discontinuity?
3. How can we formally determine the type of discontinuity using limits?
4. What happens to the function value at points of infinite discontinuity?
5. Can a function have both a removable discontinuity and an infinite discontinuity at the same point?

Tip:

Always check the left-hand and right-hand limits of the function at points of discontinuity to properly classify them.