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Oct 21 at 20:47 comment added Iosif Pinelis @SebastianGoette : This is a good point. Let me think about this.
Oct 21 at 19:49 comment added Sebastian Goette The only proof of your first formula that I know uses functions similar to $g(x)=\begin{cases}(f(x)-f(0))/x&x\ne0\\f'(0)&x=0\end{cases}$ in dimension $1$. But $g$ has one derivative less than $f$, so one cannot apply $D$ to it. But maybe you do this differently?
Oct 21 at 17:41 history answered Iosif Pinelis CC BY-SA 4.0