Talk:Semi-differentiability

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Looking at the change logs, the direction in which h approaches 0 and whether h should be added or subtracted appears to be a common source of confusion, basically leading to a revert war. Please do not make this revert again without first discussing it here and refuting my point.

If in the two expressions, both the direction of approach changes AND the operation changes, then the two expressions are equal since adding a positive h and subtracting a negative h are the same thing. So EITHER the direction of approach must be different, OR the operation must be different, but not both.

My personal preference (and this is a matter of opinion) is for the direction of approach to be different, so that the resulting expression, (f(x+h)-f(x))/h, will remind us of the standard expression used for the standard, Calculus 1 definition of a derivative (though it would be equivalent to use x-h there too, my experience is that authors seldom, if ever, write it that way). The different one-sided limits thus suggest that this is a kind of "one-sided derivative."

I hope that this will help resolve the issue and that this part of the article will not be edited by people without first reading the discussion or the change logs, but that's wishful thinking.

--Phillist (talk) 22:40, 19 March 2008 (UTC)

Another thing, shouldn't it say, the first of the limits is equal to the other, not the opposite? Then the limit (as opposed to one-sided limits) exists, and hence the derivative, which is defined to be that limit. For example, in the absolute value function, the left derivative around 0, whose value is -1, is indeed the opposite of the right derivative around 0, whose value is +1. But absolute value is certainly NOT differentiable around 0. Identity, and not additive inverses, is the correct condition.

--Phill (talk) 19:51, 20 March 2008 (UTC)

You are right. The signs got flipped around that formula a few times without anybody synchronizing with the text below. Now it is all correct. Oleg Alexandrov (talk) 19:38, 13 April 2008 (UTC)