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

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The discussion centers on the mathematical condition where if f(X+a) = f(X-a) for an infinitesimally small 'a', it leads to the conclusion that the derivative f'(X) equals 0. Participants clarify that this is true only if the function f is differentiable at X. Examples such as f(x) = |x| and f(x) = sin(x)/x illustrate cases where the condition holds but the derivative does not equal zero due to discontinuity or undefined behavior at specific points. The consensus is that continuity alone does not guarantee the derivative's existence.

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  • #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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