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!