Assume that a point [itex]x[/itex] is an interior point of domain of some function [itex]f:[a,b]\to\mathbb{R}[/itex], and assume that the limit(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\lim_{(\delta_1,\delta_2)\to (0,0)} \frac{f(x+\delta_2)-f(x+\delta_1)}{\delta_2-\delta_1}

[/tex]

exists. What does this imply?

Well I know it implies that [itex]f'(x)[/itex] exists, but does it imply more? Does it imply that the derivative of [itex]f[/itex] exists in some neighbourhood of [itex]x[/itex] and is continuous? Where's a counter example?

To be more presice, we can define a set

[tex]

\mathcal{D}=\big\{(\delta_1,\delta_2)\;\big|\; |\delta_1|<D,\; |\delta_2|<D,\; \delta_1\neq \delta_2\big\}

[/tex]

with some small [itex]D[/itex], and then consider the given expression as a mapping

[tex]

\mathcal{D}\to\mathbb{R}

[/tex]

So the domain is a square with a thin diagonal removed. Then we seek its limit at the center point [itex](0,0)[/itex].

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# Strong differentiability condition

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