Given the graph of y = f(x) in the figure, determine the following.

The x y coordinate plane is given. There is a curve and a vertical dashed line on the graph. The curve f(x) enters the window in the second quadrant, goes down and right, passes through a point above x = A, crosses the negative x-axis, sharply changes direction at a point below x = B, goes up and right, becomes nearly horizontal as it crosses the negative x-axis, goes up and right, passes through a point above x = C, passes through a point above x = D, ends at an open point above x = E, restarts at an open point below the previous open point and above x = E, goes up and right, crosses the positive y-axis, exits the window almost vertically left of the vertical dashed line at x = F, reenters the window almost vertically right of the vertical dashed line at x = F, goes down and right, changes direction at a point above x = G, goes up and right, sharply changes direction at a point above x = H, goes down and right, passes through the point above x = I, crosses the positive x-axis at x = J, and exits the window in the fourth quadrant.

(a) for which x-values A, E, G, H, or I the function is continuous (Select all that apply.)

(b) for which x-values A, E, G, H, or I the function is differentiable (Select all that apply.)

Let's analyze the function based on the description of the graph and determine continuity and differentiability at the specified points \( A \), \( E \), \( G \), \( H \), and \( I \).

### (a) **For which x-values the function is continuous?**

A function is continuous at a point \( x = a \) if:
- The function is defined at \( a \),
- The limit of the function exists as \( x \to a \),
- The limit of the function as \( x \to a \) is equal to \( f(a) \).

Based on the description of the graph:
- At \( A \), the curve is smooth and passes through the point above \( x = A \), indicating that the function is continuous at \( A \).
- At \( E \), there is an open point, which indicates a **discontinuity**. Hence, the function is not continuous at \( E \).
- At \( G \), the graph does not mention any discontinuity, and it changes direction smoothly, so the function is continuous at \( G \).
- At \( H \), the graph has a sharp change in direction, but there is no discontinuity mentioned, indicating the function is continuous at \( H \).
- At \( I \), the graph passes through smoothly, indicating the function is continuous at \( I \).

Thus, the function is continuous at \( A \), \( G \), \( H \), and \( I \), but **not continuous** at \( E \).

### (b) **For which x-values the function is differentiable?**

A function is differentiable at a point \( x = a \) if:
- The function is continuous at \( a \),
- The derivative (slope) exists, meaning the function has no sharp corners or vertical tangents at \( a \).

Based on the description of the graph:
- At \( A \), the curve is smooth, so the function is differentiable at \( A \).
- At \( E \), there is a discontinuity (open point), so the function is **not differentiable** at \( E \).
- At \( G \), the curve changes direction smoothly, so the function is differentiable at \( G \).
- At \( H \), there is a **sharp change in direction**, which means the derivative does not exist at this point. So, the function is **not differentiable** at \( H \).
- At \( I \), the curve passes smoothly, so the function is differentiable at \( I \).

Thus, the function is differentiable at \( A \), \( G \), and \( I \), but **not differentiable** at \( E \) or \( H \).

### Summary:
- **Continuity**: The function is continuous at \( A \), \( G \), \( H \), and \( I \), but not at \( E \).
- **Differentiability**: The function is differentiable at \( A \), \( G \), and \( I \), but not at \( E \) or \( H \).

Would you like any further clarifications or details?

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Here are some related questions to explore:
1. What is the formal definition of differentiability at a point?
2. How does a vertical tangent impact differentiability?
3. What types of discontinuities exist (e.g., removable, jump)?
4. Can a function be continuous but not differentiable? Provide an example.
5. How can the limit definition of a derivative be used to check differentiability?

**Tip**: A point of discontinuity cannot have a derivative, but a function can have continuous points where the derivative doesn't exist, like sharp corners or cusps!

Given the graph of y = f(x) in the figure, determine the following.
(a) for which x-values A, E, G, H, or I the function is continuous (Select all that apply.)
(b) for which x-values A, E, G, H, or I the function is differentiable (Select all that apply.)

Explore how to determine the continuity and differentiability of a function based on a graph. Analyze points A, E, G, H, and I and understand where the function is continuous or differentiable. Suitable for students in Grades 11-12.

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