If f(x+a) = f(x-a) where a is infinitesimally small then....

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Discussion Overview

The discussion revolves around the implications of the equation f(x+a) = f(x-a) where 'a' is infinitesimally small. Participants explore whether this leads to the conclusion that the derivative f'(X) equals zero, examining the conditions under which this might hold true. The conversation includes theoretical considerations and mathematical reasoning.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that if f(x+a) = f(x-a), then f'(X) must equal zero, arguing that this follows from differentiating a constant with respect to a variable.
  • Others challenge this conclusion, stating that inserting the infinitesimal 'a' into the derivative is inappropriate, as 'a' is part of the limit process in differentiation.
  • A participant questions whether the equation holds for all x and all infinitesimals a, suggesting that the result may not be universally applicable.
  • Some participants provide counterexamples, such as f(x) = |x| and f(x) = sin(x)/x, to illustrate that the conclusion does not necessarily follow in all cases.
  • There is discussion about the notation used for derivatives, with some suggesting that the notation df(a)/dx can lead to confusion regarding its interpretation.
  • A later reply indicates that if f is differentiable at X, then the claim could be true, but emphasizes that this depends on the function's properties.
  • One participant reflects on a friend's example using f(x) = x^2, arguing that the derivative at a specific point does not yield zero, despite the function appearing symmetric around that point.
  • Another participant elaborates on the calculations involving f(2-a) and f(2+a) to demonstrate that there is indeed a non-zero difference between them, raising questions about the interpretation of derivatives in this context.

Areas of Agreement / Disagreement

Participants express differing views on whether the conclusion that f'(X) = 0 follows from the initial condition. There is no consensus, as some argue in favor of the conclusion while others provide counterexamples and challenge the assumptions involved.

Contextual Notes

Limitations include the dependence on the specific function being analyzed and the potential ambiguity in the notation used for derivatives. The discussion also highlights the need for clarity regarding the conditions under which the initial equation holds.

  • #31
sgphysics said:
It is always true for a function that is continuous around x. You can not say that if it is discontinuous. But sure, you may find examples of discontinuous functions where it also holds.

What is always true for a continuous function? That ##f(x+a) = f(x-a)##? That would be false.
 
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  • #32
micromass said:
What is always true for a continuous function? That ##f(x+a) = f(x-a)##? That would be false.

Well, seeing that he is quoting my reply to #18, the logical conclusion would be that it is referring to #18 and thus saying that a continuous function is always continuous, but that seems like a tautology. I would say he is referring to ##f'(x) = 0##, but then that has already been established in this thread.
 
  • #33
micromass said:
What is always true for a continuous function? That ##f(x+a) = f(x-a)##? That would be false.
The OPs question, as I understand iit, is if limit f(x+a)=f(x-a) as a-->0. It is obviously true for a function cont. around x. Always. It is mererly the definition of a continuous function.
 
  • #34
sgphysics said:
The OPs question, as I understand iit, is if limit f(x+a)=f(x-a) as a-->0. It is obviously true for a function cont. around x. Always. It is mererly the definition of a continuous function.
This is not what the OP is asking. The OP is asking if f(X+a) = f(X-a) implies f'(X) = 0. It does if the function is differentiable at X, continuity is not sufficient as the derivative might not exist. An example which is continuous and satisfies the relation at X = 0 would be Weierstrass' function, which is not differentiable at X = 0 (or anywhere else for that matter). A more mundane example would be ##f(x) = |x|## (also at ##x = 0##), which is continuous but not differentiable at ##x = 0##.

The OP's question has been answered, I am closing therefore closing this thread.
 

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