- #31
jostpuur
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Haha. This makes me happy. I wasn't the only one making the mistake 
A question professor couldnt solve
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Discussion Overview
The discussion revolves around the question of whether it is possible to have a limit of the derivative of a function as x approaches a certain point equal to a value D, while the derivative at that point exists but is not equal to D. Participants explore examples, counterexamples, and theorems related to this concept, including the intermediate value property of derivatives.
Discussion Character
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants propose that if a function is differentiable at a point, then the limit of the derivative as x approaches that point must equal the derivative at that point.
- One participant suggests the function f(x) = (x+1)/(x+1) as an example, arguing that the limit of f'(x) as x approaches -1 is 0, while f'(-1) is undefined.
- Another participant challenges the reasoning about the limit's existence, suggesting that the limit does exist and asking for verification of their claim.
- A participant expresses doubt about whether a function can exist where the limit of f'(x) as x approaches a equals D, while f'(a) exists but is not equal to D, stating they believe this is not possible.
- Some participants reference the intermediate value property of derivatives, suggesting that it prevents the existence of such a function.
- Counterexamples are discussed, including the construction of piecewise linear functions that do not have the intermediate value property for their derivatives.
- Participants note that the original question may have been misinterpreted, leading to confusion about the existence of such functions.
Areas of Agreement / Disagreement
There is no consensus on whether such a function exists. Some participants assert that it cannot exist due to the intermediate value property, while others explore examples and counterexamples that challenge this view.
Contextual Notes
Participants reference the intermediate value property and Darboux's theorem, indicating that the discussion is heavily reliant on these mathematical concepts. There is also mention of the need for clarity in definitions and assumptions regarding differentiability and continuity.
Who May Find This Useful
This discussion may be useful for students and educators in mathematics, particularly those interested in calculus, derivatives, and the properties of functions.
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