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Differentiability: Theorems & Proofs Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
FAQs on Differentiability: Theorems & Proofs Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is the definition of differentiability? |  |
Ans. Differentiability is a property of functions that describes their smoothness and ability to be approximated by a linear function at a specific point. A function is said to be differentiable at a point if its derivative exists at that point.
| 2. What are the conditions for differentiability? | |
Ans. In order for a function to be differentiable at a point, it must satisfy two conditions: (1) the function must be continuous at that point, and (2) the limit of the difference quotient, which represents the derivative, must exist.
| 3. What is the differentiability theorem? | |
Ans. The differentiability theorem states that if a function is differentiable at a point, then it must be continuous at that point. In other words, differentiability implies continuity.
| 4. How can the differentiability of a function be proved? | |
Ans. The differentiability of a function can be proved using the definition of the derivative. By showing that the limit of the difference quotient exists as the independent variable approaches a specific point, one can establish the differentiability of the function at that point.
| 5. Are all continuous functions differentiable? | |
Ans. No, not all continuous functions are differentiable. While differentiability implies continuity, the converse is not true. There exist functions that are continuous at a point but fail to have a derivative at that point. An example of such a function is the absolute value function at x=0.
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