The four Dini derivatives are: $$ D^+F(x) = \limsup_{h\rightarrow 0^+}\frac{F(x+h)-F(x)}{h}, \\ D^-F(x) = \limsup_{h\rightarrow 0^-}\frac{F(x+h)-F(x)}{h} = \limsup_{h\rightarrow 0^+}\frac{F(x)-F(x-h)}{h}, \\ D_+F(x) = \liminf_{h\rightarrow 0^+}\frac{F(x+h)-F(x)}{h}, \\ D_-F(x) = \liminf_{h\rightarrow 0^-}\frac{F(x+h)-F(x)}{h} = \liminf_{h\rightarrow 0^+}\frac{F(x)-F(x-h)}{h}$$ Now I wonder what is the $D^+(-F(-x))$? My instructor claimed that $D^+(-F(-x)) = D^-F(x)$, but I think $D^+(-F(-x)) = D^-F(-x)$. So which is correct? This troubles me and I would appreciate any help.
2026-04-03 22:07:06.1775254026
I want to raise a question about the notation of Dini derivatives
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