From the graph you uploaded, you are being asked to determine the discontinuities in the function $f(x)$ at various $x$-values. For each discontinuity, you need to classify the type of continuity:

1. Continuous from the left: The function approaches the same value from the left as it is at the point.
2. Continuous from the right: The function approaches the same value from the right as it is at the point.
3. Neither: The function is discontinuous from both directions (left and right).

Let's analyze the points of discontinuity:

1. At $x = -4$: The function has an open circle, meaning that it does not have a value at $x = -4$, and there is no continuity from either the left or the right. This is a discontinuity where neither side is continuous.

2. At $x = -2$: The graph has a filled point on the left and an open circle on the right. This suggests that the function is continuous from the left, but not from the right.

3. At $x = 0$: There is no apparent discontinuity at $x = 0$, so the function is continuous at this point.

4. At $x = 2$: The graph has an open circle, suggesting a discontinuity. However, the function approaches a value from both sides, but it does not exist at $x = 2$. This is a discontinuity where neither the left nor the right side is continuous.

5. At $x = 4$: The graph drops down, showing a jump discontinuity. The function is continuous from the left but not from the right.

6. At $x = 6$: The graph stops at $x = 6$ with an open circle, indicating a discontinuity. It is discontinuous from both sides, so neither side is continuous here.

Summary of discontinuities:

• $x = -4$: Neither
• $x = -2$: Continuous from the left
• $x = 2$: Neither
• $x = 4$: Continuous from the left
• $x = 6$: Neither

Would you like more details on how to classify types of discontinuities or have any questions about the specific points on the graph?

Follow-up Questions:

1. What are the conditions for a function to be continuous at a point?
2. How is a "removable discontinuity" different from a "jump discontinuity"?
3. What does it mean for a function to be continuous from the right or left?
4. How can discontinuities affect the limit of a function at a given point?
5. What other types of discontinuities exist besides "jump" and "removable"?

Tip:

For a function to be continuous at a point, the left-hand limit, right-hand limit, and the function's value at the point must all be equal.