Let's analyze the graph to determine the types of discontinuities at each $x$-value.

1. At $x = -4$:

   • From the graph, it appears that the function has a jump discontinuity at $x = -4$. There are two distinct parts of the graph on either side of $x = -4$, indicating that the left-hand and right-hand limits do not match.

2. At $x = -2$:

   • At this point, the graph shows a removable discontinuity because there is a hole at $x = -2$, but the graph approaches the same value from both sides. The function is not defined at this point, but the limits from the left and right are equal.

3. At $x = 2$:

   • The graph suggests a removable discontinuity here as well. There is a hole at $x = 2$, but the function could be redefined to fill the hole, as the left-hand and right-hand limits match.

4. At $x = 4$:

   • The graph seems to exhibit a jump discontinuity at $x = 4$, similar to the case at $x = -4$. The left-hand and right-hand limits do not approach the same value, and there is a jump between two parts of the function.

If you'd like more details on any of these, feel free to ask!

Here are some follow-up questions for deeper understanding:

1. How can you formally define the different types of discontinuities?
2. What is the difference between a removable discontinuity and a jump discontinuity?
3. Can a function be continuous at a point where it has a removable discontinuity if redefined?
4. How do limits from the left-hand side and right-hand side relate to determining the type of discontinuity?
5. How would you find discontinuities algebraically, rather than just graphically?

Tip: Always check if the left-hand and right-hand limits are equal when identifying discontinuities!