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Could someone please give me an example of two functions $f$ and $g$ such that $$D^+(f+g)\neq D^+(f)+D^+(g)$$ where $D^+f$ is the upper right derivative of $f$ (i.e. $D^+ f(x) = \limsup \limits_{h\to 0^+} \frac{f(x+h) - f(x)}{h}$)?

I would appreciate any help. Thanks.

Sam
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  • What exactly are you looking for? You could come up with many examples yourself just by picking $f$ and $g$ to be any differentiable functions. – Philip Hoskins Mar 04 '17 at 22:23
  • @ Philip Hoskins Ohh sorry, I was to mean not equal. I edited it now. – Sam Mar 05 '17 at 09:19

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Take two functions $f,g:\mathbb{R}\to\mathbb{R}$ defined by $$ f(x)=\begin{cases}x&\text{if }x\in\mathbb{Q}\\ -x&\text{if }x\notin\mathbb{Q} \end{cases},\qquad g(x)=-f(x). $$ Then you can verify that $D^+(f+g)(0)=0$ and $D^+f(0)+D^+g(0)=2$.

choco_addicted
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  • Are you able to help out with this question of mine? :) https://math.stackexchange.com/questions/2953842/proving-an-inequality-for-upper-right-hand-dini-derivatives – Homaniac Oct 13 '18 at 19:46