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.