https://file.aiquickdraw.com/mathbot/file/17265286012170c0udjz0.jpg

Let's analyze the graph of \( f(x) \) from the image to answer the questions.

### Part A: Identify all the x-values in the interval (-4, 5) where the graph of \( f \) is **not continuous**.

A function is not continuous at a point if there is a break, jump, or hole in the graph at that point. 

From the graph:
- At \( x = -1 \), there is a **hole**, indicating that \( f \) is not continuous at this point.
- At \( x = 1 \), there is a **jump discontinuity**, so the graph is not continuous here as well.

Thus, the x-values where the graph is not continuous are:
\[
x = -1, \quad x = 1
\]

### Part B: Identify all the x-values in the interval (-4, 5) where the graph of \( f \) is **not differentiable**.

A function is not differentiable at a point where:
1. The function is not continuous (since differentiability implies continuity).
2. The graph has a sharp corner or cusp.
3. There is a vertical tangent.

From the graph:
- At \( x = -1 \) and \( x = 1 \), since the graph is not continuous, it is also not differentiable.
- At \( x = 0 \), there is a **sharp corner**, so the function is not differentiable at this point as well.

Thus, the x-values where the graph is not differentiable are:
\[
x = -1, \quad x = 0, \quad x = 1
\]

Would you like more details on this or any further clarifications?

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Here are 5 related questions to deepen your understanding:
1. What is the formal definition of continuity at a point?
2. How does differentiability relate to continuity?
3. Can a function be continuous but not differentiable? Provide an example.
4. What are the different types of discontinuities a function can have?
5. What is the geometric meaning of a "sharp corner" in a graph?

**Tip**: A function is always differentiable where its graph is smooth and continuous, but keep an eye on points with sharp turns or jumps—those are places of non-differentiability!

Identify all the x-values in the interval (-4, 5) where the graph of f is not continuous. Identify all the x-values in the interval (-4, 5) where the graph of f is not differentiable.

Learn how to identify points of discontinuity and non-differentiability on the graph of f(x) in the interval (-4, 5). This problem covers key concepts of continuity, differentiability, and the behavior of functions at sharp corners and jump discontinuities. Suitable for high school students.

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