The graph provided shows a piecewise function with various sections of line segments, suggesting potential discontinuities.

Step-by-step analysis:

1. Left-side segment:

   • For $x < -2$, the function appears continuous, following a straight line. There is no apparent discontinuity for $x < -2$.

2. Point at $x = -2$:

   • The graph shows a change in direction at $x = -2$, but no jump or break. The left and right-hand limits as $x \to -2$ seem to coincide with the function value at that point. Thus, the function is continuous at $x = -2$.

3. Point at $x = 0$:

   • There is a break at $x = 0$. On the left side of $x = 0$, the function follows a downward sloping line, while on the right side, the function shifts upward (horizontal line) starting at a different height. Since the left-hand and right-hand limits as $x \to 0$ do not match, the function is discontinuous at $x = 0$.

4. Point at $x = 1$:

   • At $x = 1$, the graph shows a solid dot at a specific height, but the function continues as a horizontal line beyond this point. There seems to be no jump or gap here, implying continuity at $x = 1$.

Conclusion:

The function is discontinuous at $x = 0$.

Would you like more details on this discontinuity or any other questions?

Here are some related questions you might find interesting:

1. What are the types of discontinuities, and which one is at $x = 0$?
2. How can limits help determine continuity?
3. Could a function be continuous but not differentiable?
4. How would the graph change if the discontinuity at $x = 0$ were removed?
5. Can a piecewise function always be made continuous?

Tip: When analyzing graphs for continuity, always check the left-hand and right-hand limits, as well as the function's value at the point.