What can be inferred about the relationship between the function's continuity and its differentiability based on the passage?
Practice Questions
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Q1
What can be inferred about the relationship between the function's continuity and its differentiability based on the passage?
Continuity implies differentiability.
Differentiability implies continuity.
Both are independent properties.
Neither is necessary for the other.
Differentiability at a point implies that the function is continuous at that point, but continuity does not guarantee differentiability.
Questions & Step-by-step Solutions
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Q
Q: What can be inferred about the relationship between the function's continuity and its differentiability based on the passage?
Solution: Differentiability at a point implies that the function is continuous at that point, but continuity does not guarantee differentiability.
Steps: 4
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.
