Does the existence of a limit at x=0 prove the existence of f'(0)?

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Discussion Overview

The discussion centers on the relationship between the existence of a limit of the derivative of a function as it approaches zero and the existence of the derivative at that point. The context involves theoretical reasoning about calculus, particularly focusing on differentiability and continuity of functions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant proposes that if f is continuous and differentiable for all nonzero x, and if the limit of f'(x) as x approaches 0 exists, then it should imply that f'(0) exists.
  • Another participant suggests using the Mean Value Theorem to support the argument regarding the existence of f'(0).
  • A third participant provides LaTeX formatting tips for mathematical expressions, indicating a focus on clarity in mathematical communication.
  • One participant notes that while derivatives are not necessarily continuous, they possess the intermediate value property, which could be relevant to the discussion on limits and derivatives.
  • This participant further argues that if the limit of f'(x) exists as x approaches a, then f'(a) must also exist and is equal to that limit.

Areas of Agreement / Disagreement

The discussion includes multiple viewpoints and approaches, with no clear consensus reached on whether the existence of the limit at x=0 definitively proves the existence of f'(0).

Contextual Notes

Participants have not fully resolved the implications of the Mean Value Theorem or the intermediate value property in relation to the existence of derivatives at specific points.

Demon117
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I wanted to see what kind of responses I would get regarding this problem:

Let f : \Re\rightarrow\Re be a continuous function that is differentiable for all nonzero x such that f '(x) exists. If f'(x) \rightarrow L as x\rightarrow0 exists, prove that f '(0) exists.
 
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Use the Mean Value Theorem
 
A few LaTeX tips: Put tex or itex tags around the whole formula instead of around each symbol, and use \mathbb R for the set of real numbers. Example: f:\mathbb R\rightarrow\mathbb R. (Click the quote button to see what I did).
 
While derivatives are not necessarily continuous, they do satisfy the "intermediate value property" (if f'(a)< c< f'(b), then f'(d)= c for some d between a and b). You can show that using the mean value theorem JG89 suggests. From that it follows that if \displaytype\lim_{x\to a}f&#039;(x) exists, then so does f'(a) and \displaytype f&#039;(a)= \lim_{x\to a}f&#039;(x)
 

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