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No. There are functions that are continuous but aren't differentiable. For example is the function where it looks like a big "V" when graphed. I don't know the theoretical explanation but that's the best I got :D
the theorem states that if you can take the derivative of a function it implies that it is continuous.
If it is differentiable, then it is continuous. A continuous function is not always differentiable...but that was not the question.
|dw:1328658896786:dw| If a function is continuous it does not implie that it is differentiable. the absolute function is an example of this because you can not take the derivative at the corner.
the answer is yes.
Yes, but if you are studying AP calculus (students tend to think that the theorem is an iff case.
You are correct, they tend to think continuity implies differentiability....not true as you stated with your example...cusps would be another good example used in an ap calculus class.
yes besides the whole point of this website is to communicate about mathematics. I am always amaze how some people are very good at solving problems in different ways.