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FAQs on Continuity And Differentiability - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is the definition of continuity in mathematics? |  |
| 2. How do you determine if a function is continuous at a specific point? | |
Ans. To determine if a function is continuous at a specific point, we need to check three conditions:
1) The function must be defined at that point.
2) The limit of the function as x approaches the given point must exist.
3) The value of the function at the given point must be equal to the limit.
| 3. What is the difference between continuity and differentiability? | |
Ans. Continuity and differentiability are related concepts but have distinct meanings. Continuity refers to the smoothness and connectedness of a function's graph, whereas differentiability focuses on the existence of the derivative of a function at a particular point. A function can be continuous but not differentiable, but if a function is differentiable, it is always continuous.
| 4. Can a function be differentiable but not continuous? | |
Ans. No, a function cannot be differentiable but not continuous. Differentiability implies continuity. If a function is differentiable at a point, it must be continuous at that point as well. Discontinuities in a function prevent it from being differentiable.
| 5. Are all continuous functions differentiable? | |
Ans. No, not all continuous functions are differentiable. While continuity is a necessary condition for differentiability, it is not sufficient. A function must also have a well-defined tangent line at each point for it to be differentiable. Functions with sharp corners or vertical tangents, for example, are continuous but not differentiable at those points.
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