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Determine if the function below is continuous.

The graph provided shows a piecewise function with two linear segments, both of which intersect at a point. The primary focus for determining continuity is the point where the segments meet, especially any potential discontinuity.

### Analysis:
1. **Left segment:** The left segment descends toward a vertex at \( x = 1 \).
2. **Right segment:** The right segment ascends from the same vertex at \( x = 1 \).
3. **Vertex at \( x = 1 \):** There is an open circle at the point \( (1, -1) \) and a filled circle at the point \( (1, -2) \).

### Continuity Check at \( x = 1 \):
For the function to be continuous at \( x = 1 \):
- The left-hand limit as \( x \) approaches 1 must equal the right-hand limit.
- The value of the function at \( x = 1 \) must equal these limits.

#### Observations:
- The left-hand limit as \( x \) approaches 1 is \( -1 \) (from the open circle).
- The right-hand limit as \( x \) approaches 1 is \( -1 \) (from the open circle).
- The actual value of the function at \( x = 1 \) is \( -2 \) (since the filled circle is at \( (1, -2) \)).

Since the limit from both sides does not equal the actual value of the function at \( x = 1 \), the function is **not continuous** at this point.

### Conclusion:
The function is **discontinuous** at \( x = 1 \).

Do you have any further questions or would you like a more detailed explanation?

### Related Questions:
1. What types of discontinuities exist in a function?
2. How can a piecewise function be made continuous?
3. What are the mathematical conditions for a function to be continuous at a point?
4. Can a function be continuous if it has an open circle at a point?
5. How does differentiability relate to continuity?

### Tip:
For any function to be continuous at a point, make sure the left-hand limit, right-hand limit, and the function value at that point all coincide.

Understand how to determine the continuity of a piecewise function with a detailed analysis. This problem involves assessing continuity at a specific point using limits and function values. Suitable for high school students studying calculus and functions.

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