The graph you've provided shows a piecewise function that is mostly continuous but has specific points that could raise concerns about differentiability.

In general, a function is differentiable at a point if:

1. It is continuous at that point (no jumps, gaps, or holes).
2. There is no sharp corner or cusp at that point.
3. The slope of the tangent line exists and is finite.

Observations:

• At $x = -3$: The graph is continuous and smooth, meaning the function is likely differentiable at this point.
• At $x = -2$: There is a sharp corner, which means the function is not differentiable at this point.
• At $x = -1$: The graph has a minimum, but it looks smooth, so it is likely differentiable at this point.
• At $x = 0$: There is an open circle (a removable discontinuity), which means the function is not continuous here, so it is not differentiable.
• At $x = 1$: The graph appears smooth, so the function is likely differentiable at this point.
• At $x = 2$: There is another sharp point, meaning the function is not differentiable at this point.
• At $x = 3$: The graph appears smooth again, so the function is likely differentiable here.

Conclusion:

The function is not differentiable at $x = -2$, $x = 0$, and $x = 2$, due to a sharp corner and discontinuities. It seems differentiable at all other points.

Would you like further details on any specific intervals or have any questions?

Here are 5 related questions to explore:

1. What is the definition of differentiability in terms of limits?
2. How can we identify points of non-differentiability on a graph?
3. What role do continuity and smoothness play in differentiability?
4. How do open circles in a graph affect the function's continuity and differentiability?
5. What is the difference between a sharp corner and a smooth curve in terms of derivatives?

Tip: To check if a function is differentiable at a point, ensure both continuity and a smooth (non-sharp) transition around that point.