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What is this derivative called: $$ D_\gamma f = \lim_{dt\to 0} \frac{f(\gamma(dt))-f(\gamma(0))}{||\gamma(dt)-\gamma(0)||} $$ It is not the same as the directional derivative or the Gateux derivative on wikipedia. Perhaps it is not as general since it uses a vector norm. (or perhaps it is but I don't see it)

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  • $\begingroup$ Is there some reason you are interested in this derivative? That limit will very rarely exist... $\endgroup$ – Eric Wofsey Jun 29 '18 at 6:37
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    $\begingroup$ Are there hypothesis on $\gamma$ ? $\endgroup$ – nicomezi Jun 29 '18 at 6:38
  • $\begingroup$ (Notice, for instance, that if $\gamma(t)=t$, then $D_\gamma f$ will not exist for any ordinarily differentiable function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(0)\neq 0$.) $\endgroup$ – Eric Wofsey Jun 29 '18 at 6:40
  • $\begingroup$ @EricWofsey: Can you explain? For me, $\gamma(t) = t$ results in $D_\gamma f = \lim_{h\searrow0}\frac{f(h) - f(0)}{h - 0} = f'(0)$. $\endgroup$ – gerw Jun 29 '18 at 6:50
  • $\begingroup$ Notice the norm at the denominator. @gerw $\endgroup$ – nicomezi Jun 29 '18 at 6:52
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In the case that $\gamma$ is differentiable at $0$, this is very similar to the definition of the Hadamard derivative. There, the existence of $$\frac{f(\bar x + t_n \, h_n) - f(\bar x)}{t_n}$$ is required for all sequences $t_n \searrow 0$ and $h_n \to h$, see https://en.wikipedia.org/wiki/Hadamard_derivative.

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