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Proof of theorem stating Differentiability implies Continuity Video Lecture | Mathematics (Maths) Class 12 - JEE

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FAQs on Proof of theorem stating Differentiability implies Continuity Video Lecture - Mathematics (Maths) Class 12 - JEE

1. What is the theorem that states differentiability implies continuity?
Ans. The theorem that states differentiability implies continuity is a fundamental result in calculus. It states that if a function is differentiable at a point, then it must also be continuous at that point.
2. How can we prove the theorem that differentiability implies continuity?
Ans. The proof of the theorem can be done using the definition of differentiability and the limit definition of continuity. By showing that the limit of the difference quotient (the derivative) exists, we can conclude that the function is both differentiable and continuous at that point.
3. What does it mean for a function to be differentiable at a point?
Ans. A function is said to be differentiable at a point if its derivative exists at that point. Geometrically, this means that the function has a well-defined tangent line at that point, and the graph of the function does not have any sharp corners or cusps.
4. Can a function be continuous but not differentiable?
Ans. Yes, it is possible for a function to be continuous but not differentiable. A common example is a function with a sharp corner or a cusp at a certain point. Although the function may be continuous at that point, the derivative does not exist, and hence, the function is not differentiable.
5. Are all continuous functions differentiable?
Ans. No, not all continuous functions are differentiable. While it is true that differentiability implies continuity, the converse is not always true. There are continuous functions that do not have a well-defined derivative at certain points, such as functions with corners, cusps, or vertical tangent lines.
204 videos|288 docs|139 tests

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