What is the intuition behind the notion of strong differentiation?

Asked Oct 07 '18 at 05:21

Active Oct 07 '18 at 05:21

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${\bf definition.}$ A function $f$ defined on open set $A \subseteq \mathbb{R}$ is said to be ${\bf strong \; differentiable}$ at $a \in A$ if

$$ \lim_{(x_1,x_2) \to (a,a) \\ x_1 \neq x_2 } \dfrac{ f(x_1)-f(x_2) }{x_1-x_2} = f^{*}(a)$$

exists and is finite and we call $f^*(a)$ the strong derivative of $f$ at $x=a$

How is this notion different than the usual derivative? I mean, if we set $x_2 = a$, we get our usual derivative...

What is the point of such definition? what is motivation for this?

asked Oct 07 '18 at 05:21

James

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See “Strong” derivative of a monotone function *AND* What does the continuity of $f'$ tell us about $f$? *AND* Differences between derivatives and strong derivatives *AND* Derivatives as defined by a two variable difference quotient limit *AND* Alternative definition of derivative. – Dave L. Renfro Oct 07 '18 at 08:20
In particular if $f$ is differentiable on $B$ where $B$ is open and $a\in B\subset A$ then by the MVT, $f$ is strong differentiable at $a$ iff $f'(x) $ is continuous at $x=a.$ – DanielWainfleet Oct 07 '18 at 10:31

What is the intuition behind the notion of strong differentiation?

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