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Lower Dini derivative of function F at point x in direction d is defined by
$$DF(x):=\liminf\limits_{h\to 0^+}\frac{F(x+hd)-F(x)}{h}.$$
Consider two functions F and G, we have
$$D(F+G)(x) \geq DF(x)+DG(x). \quad(1)$$

Question: if G is differentiable, is the following true? $$D(F+G)(x) = DF(x)+DG(x). \quad(2) $$

Answer: Yes

Proof: Assume $DG(x) \to g(x)$ since $G$ is differentiable. Using (1) we have $$D(F+G)(x) \geq DF(x)+DG(x)= D(F(x)) + g(x). \quad (3)$$ Furthermore, using $ D(-G(x))=-g(x)$, we have $$D(F(x)) = D((F(x)+G(x)) +(- G(x)))$$ $$ \geq D(F(x)+G(x)) + D(-G(x)) $$ $$ =D(F(x)+G(x)) - g(x).$$ Hence $$D(F+G)(x) \leq D(F(x)) + g(x) \quad(4).$$ Combing (3) and (4), we have $$D(F+G)(x) = DF(x)+g(x).$$

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    $\begingroup$ The guy is called Dini... $\endgroup$
    – user251257
    Apr 22, 2018 at 20:14
  • $\begingroup$ Only G is differentiable. If both are differentiable, my understanding is that the equality holds. $\endgroup$
    – jsmath
    Apr 22, 2018 at 20:18
  • $\begingroup$ No, I haven't tried anything - just curious. $\endgroup$
    – jsmath
    Apr 22, 2018 at 20:21
  • $\begingroup$ I answered my own question. $\endgroup$
    – jsmath
    Apr 23, 2018 at 16:21

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