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I imagine this is basic analysis material which someone who isn't me knows off the top of their head:

Given a function $f$, we may define the difference quotient $q(x, y) = \frac{f(y) - f(x)}{y - x}$. The derivative of $f$ at some point $x$ is of course standardly defined as $\lim_{y \to x} q(x, y)$, but I'm interested in the existence of the stronger limit $\lim_{(a, b) \to (x, x), a \neq b} q(a, b)$. Let us say this limit defines the "strong derivative" of $f$ where it exists. Of course, if $f$ has a strong derivative at a point, then it also has a derivative at that point, and they match. But there are cases of functions which are differentiable at a point but not strongly differentiable (for example, the standard pathological $x^2 \sin(1/x)$ at the origin).

Wherever a function is continuously differentiable, it is also strongly differentiable. I am curious as to whether this sufficient condition is also necessary, and, if not, whether any other nice characterization can be given for strong differentiability. And, in general, I am curious as to the extent to which theorems and curiosities about derivatives carry over or don't carry over to strong derivatives.

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I posted a lot of results and references relating to the strong derivative in my answer to “Strong” derivative of a monotone function. –  Dave L. Renfro Sep 5 '12 at 17:13

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I quote a nice theorem from Bourbaki, Fonctions d'une variable réelle.

Proposition 6. Let $I$ be an open interval and let $x_0$ be one of the two extremum points of $I$. Let $f$ be a vector function defined and continuous on $I$, whose codomain is a complete normed vector space $E$. We suppose that $f$ has a right derivative $f_d'$ at each point of the complementary set $B$ of a countable subset of $I$. A necessary and sufficient condition for $f_d'(x)$ to have a limit as $x \to x_0$ in $B$ is that $$\frac{f(y)-f(x)}{y-x}$$ has a finite limit $c$ as $(x,y)$ tend to $(x_0,x_0)$ with $x$, $y\in I$, $x\neq y$.

This statement is in the chapter Derivés of the French edition. It is not a complete answer to your question, but I hope you can read the proof, which has some interesting ideas that might help you. Please notice that Bourbaki's style is often too general, and you may try to relax some condition of the previous proposition.

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