What can be inferred about the relationship between the function's continuity an

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Question: What can be inferred about the relationship between the function\'s continuity and its differentiability based on the passage?

Options:

Correct Answer: Differentiability implies continuity.

Solution:

Differentiability at a point implies that the function is continuous at that point, but continuity does not guarantee differentiability.

What can be inferred about the relationship between the function's continuity and its differentiability based on the passage?

What can be inferred about the relationship between the function's continuity and its differentiability based on the passage?

Step 1: Understand what continuity means. A function is continuous at a point if you can draw it without lifting your pencil.
Step 2: Understand what differentiability means. A function is differentiable at a point if it has a defined slope (or tangent) at that point.
Step 3: Recognize the relationship: If a function is differentiable at a point, it must also be continuous at that point. This means you can draw it smoothly there.
Step 4: However, just because a function is continuous at a point does not mean it is differentiable there. There can be points where the function is continuous but has a sharp corner or cusp, making it not differentiable.

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