To solve the problem shown in the image, we are asked to determine the $x$-values where the function $f(x)$ is discontinuous and identify the type of discontinuity (from the left, from the right, or neither) based on the graph.

Key observations from the graph:

1. The function appears to be discontinuous at certain points where there are jumps, breaks, or holes in the graph.
2. We need to find those points, then determine whether the function is continuous from the right, the left, or neither.

Analysis:

• Smallest discontinuity:

  • At $x = -2$, there is a hole in the graph, indicating a discontinuity. The function is continuous from the left but not from the right (since the function value does not exist for $x > -2$).

• Next discontinuity:

  • At $x = 2$, there is another jump in the graph. The function is continuous from the left but discontinuous from the right (since there is a gap at $x = 2$).

• Next discontinuity:

  • At $x = 4$, there is a vertical break, meaning there is no continuity from either direction (neither continuous from the left nor the right).

• Largest discontinuity:

  • At $x = 6$, there is a break again, so neither the left nor right continuity exists at this point.

Final Answers:

• Smallest value $x = -2$: Continuous from the left.
• Next value $x = 2$: Continuous from the left.
• Next value $x = 4$: Neither.
• Largest value $x = 6$: Neither.

Would you like more details on how to classify discontinuities or have any questions?

Here are five related questions to expand on this:

1. What are the different types of discontinuities (removable, jump, and infinite)?
2. How does the left-hand limit and right-hand limit determine continuity at a point?
3. What is the difference between continuity and differentiability at a point?
4. How do we check for removable discontinuities from a function's equation?
5. Can a function be discontinuous at more than one point and still be integrable?

Tip: A function is continuous at a point if the left-hand limit, right-hand limit, and function value all exist and are equal.