Let's analyze the graph to answer the given questions.

Part (a): Find all values for which the function is discontinuous.

The function appears to be discontinuous at the following points:

• $x = -1$: There is a hole in the graph at this point.
• $x = 1$: There is a jump discontinuity, with a gap between two distinct function values.
• $x = 2$: The graph has a hole at this point.

So, the function is discontinuous at $x = -1$, $x = 1$, and $x = 2$.

Part (b): For each value in part (a), state why the formal definition of continuity does not apply.

1. At $x = -1$:

   • The limit exists as $x$ approaches -1, but the function is not defined at $x = -1$ (there is a hole), meaning $f(-1)$ is not defined.

2. At $x = 1$:

   • The left-hand limit and the right-hand limit exist but are not equal. This is a jump discontinuity because $\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)$.

3. At $x = 2$:

   • There is a hole in the graph at $x = 2$, which implies that while the limit exists, the function is not defined at $x = 2$.

Part (c): Classify each discontinuity as either jump, removable, or infinite.

1. At $x = -1$:

   • This is a removable discontinuity since the limit exists, but the function value is not defined.

2. At $x = 1$:

   • This is a jump discontinuity because the left-hand and right-hand limits are not equal, creating a gap.

3. At $x = 2$:

   • This is a removable discontinuity since there is a hole at $x = 2$ (the limit exists but the function is not defined).

Would you like further clarification on any of these points?

Here are five related questions for practice:

1. How would the graph of the function look if the discontinuities were removed?
2. What is the formal definition of a continuous function at a point?
3. Can a function have a removable discontinuity and still be differentiable at that point? Why or why not?
4. How does a jump discontinuity affect the integrability of a function?
5. What are the implications of discontinuities on the behavior of the derivative at those points?

Tip:

When classifying discontinuities, always check the behavior of the function's limit at the point in question as well as whether the function is defined there.