https://file.aiquickdraw.com/mathbot/file/172705436509304kjqq2m.jpg

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 \( x = 2 \).

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### 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 \( x = 2 \).

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### 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).

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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.

Consider the graph of the function y = f(x) shown in the following graph. (See attached image). a. Find all values for which the function is discontinuous. b. For each value in part a., state why the formal definition of continuity does not apply. c. Classify each discontinuity as either jump, removable, or infinite.

Analyze the graph of the function to identify discontinuities at x = -1, x = 1, and x = 2. Learn why the formal definition of continuity does not apply at these points and classify the types of discontinuities (jump and removable). Suitable for students in Grades 10-12 studying limits and continuity.

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