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Determine the intervals over which the function shown below is continuous.

Enter your answer in interval notation. Use a comma to separate multiple intervals.

To determine the intervals where the given function is continuous, I will analyze the graph. From what I can see, the function has the following key features:

1. **Continuous segments**: The function appears to be continuous at all points except one specific location.
2. **Discontinuity at \( x = -2 \)**: The vertical dashed red line at \( x = -2 \) indicates a vertical asymptote or a discontinuity.
3. **Open circle at \( x = -6 \) and \( x = 4 \)**: This suggests the function is not defined at these points, leading to discontinuities.

Therefore, the function is continuous in the following intervals:
- From \( -\infty \) to \( -6 \)
- From \( -6 \) to \( -2 \) (as there is a discontinuity at \( x = -2 \))
- From \( -2 \) to \( 4 \)
- From \( 4 \) to \( \infty \)

### Answer:
The function is continuous on the intervals:
\[
(-\infty, -6) \cup (-6, -2) \cup (-2, 4) \cup (4, \infty)
\]

Would you like more details or have any further questions? Here are some related questions to explore:

1. What types of discontinuities can occur in a function's graph?
2. How do you determine if a function has a removable discontinuity?
3. Can a function be differentiable if it is not continuous?
4. What is the formal definition of a continuous function at a point?
5. How does the Intermediate Value Theorem relate to the continuity of a function?

**Tip**: When analyzing graphs for continuity, always look for vertical asymptotes, holes (open circles), and jumps in the graph as indicators of discontinuities.

Determine the intervals over which the function shown below is continuous. Enter your answer in interval notation. Use a comma to separate multiple intervals.

Learn how to determine the intervals of continuity for a function using a given graph. This problem focuses on identifying discontinuities, vertical asymptotes, and open circles in the graph. Suitable for students in Grades 11-12 or college-level calculus.

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