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Most (all?) textbooks on calculus state unconditionally that a function f is differentiable at x if and only if both of the one-sided derivatives $f_+'(x)$ and $f_-'(x)$ exist and are equal. On the other hand, answers to this question implies that that is not the case. So, are the textbooks or answers to that question wrong?

M. Nejat Aydin
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    Can you please state one example of a Calculus textbook that claims that? – José Carlos Santos Mar 01 '24 at 15:36
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    The question you cited deals with a function defined only to the right side of the point, and the theorem you cite usually (always?) requires that the function be defined at an interior point of the domain. When the function is not defined at an interior point of the domain, then authors will vary on what it means to be differentiable at the point. – Dave L. Renfro Mar 01 '24 at 15:43
  • @JoséCarlosSantos "Theory and Problems of Advanced Calculus, M.R.Spiegel" – M. Nejat Aydin Mar 01 '24 at 15:43
  • @DaveL.Renfro The problem is that most (all?) authors neglect to state that the theorem is valid only for an interior point of the domain. – M. Nejat Aydin Mar 01 '24 at 15:49
  • most (all?) authors neglect --- This might be the case for elementary calculus texts, but probably not for upper (undergraduate) level real analysis texts. Also, it's possible that in some of the texts there had been an earlier stipulation that only functions defined on an open set are considered. In any event, this is not something to be especially concerned with, unless you're writing your own text or you are proofreading another person's text. To me this seems something along the lines of is a square a rectangle -- you'd think YES, but it's amazing how often actual usage suggests NO. – Dave L. Renfro Mar 01 '24 at 16:07
  • Regarding my last comment, see page 31 of this document. – Dave L. Renfro Mar 01 '24 at 16:09
  • @DaveL.Renfro Thanks. Could you name an upper level text that deals properly with this issue? I am curious about the wording of the theorem. – M. Nejat Aydin Mar 01 '24 at 16:23
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    See pp. 398-399 of Elementary Real Analysis by Bruckner/Bruckner/Thomson (2008, 2nd edition). – Dave L. Renfro Mar 01 '24 at 16:42
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    WARNING: Make sure you understand that the right-hand derivative $f'+(x)$ is defined as a difference quotient (with $h>0$) and not as $\lim{y\to x^+} f'(y)$. A lot of students mess this up. – Ted Shifrin Mar 01 '24 at 20:30

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If you take a function: $$ f( x) =\begin{cases} x^{2} & ,x\leqslant 0\\ x & ,x >0 \end{cases} $$ Then at $x=0$ we can't agree on what the derivative is because there are 2 choices. Which one would you choose - the left or the right derivative? That's why the (two-sided) derivative doesn't exist at that point.

But if the whole function is:

$$ g(x) = x^{2} ,x\leqslant 0 $$

Then there's no ambiguity at $x=0$, there's only one derivative there. So at end points of a function (or at other points where the domain is interrupted) we can still say that the derivative exists. It's just most of the time text books don't talk about the endpoints.

Stanislav Bashkyrtsev
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