To determine the intervals where $f(x)$ is continuous based on the graph, let's examine the function:

• Continuity requires that the graph has no breaks, jumps, or holes.
• Based on the graph provided, there are no breaks between $x = -7$ and $x = 4$ except at two points:
  1. There is a hole at $x = -6$, indicating discontinuity there.
  2. At $x = 2$, the graph has a sharp break, indicating discontinuity as well.

Conclusion:

• The function is continuous in the following intervals (excluding the points of discontinuity at $x = -6$ and $x = 2$): $(-7, -6) \cup (-6, 2) \cup (2, 4]$

Would you like further details on this solution, or do you have any other questions?

Here are 5 related questions to further your understanding:

1. What types of discontinuities exist, and how do they differ?
2. How can you tell if a function has a removable discontinuity from its graph?
3. How do we formally define continuity at a point using limits?
4. How would the intervals change if the graph had a vertical asymptote?
5. How can you determine whether a point of discontinuity is removable or non-removable?

Tip: When checking continuity from a graph, always look for holes, jumps, and vertical asymptotes, as these indicate points where the function is not continuous.