# Derivatives as defined by a two variable difference quotient limit

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

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 $x$, $y\in I$, $x\neq y$.