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$$f \left(\frac{x+y}{2} \right) = \frac{f(x)+f(y)}{2}, \forall x,y \in \mathbb{R}$$

$$f'(0)= -1,\space f(0)=1$$

$$f'(u)=?$$

Vivek Kaushik
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Aakif Akhtar
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  • Just differentiate the functional equation with respect to $y$ and put $y=0$. – Paramanand Singh Jul 01 '17 at 15:30
  • Well, the first step should be to prove that it's differentiable at all, and not only for $x=0$. But why do you need that, is it homework, or something important? –  Jul 01 '17 at 15:41

1 Answers1

1

Hint

Prove first that $f(-x)=2-f(x)$. Then, $$\frac{f(x+h)-f(x)}{h}=\frac{f(x+h)+f(-x)-2}{h}=\frac{2f(\frac{h}{2})-2}{h}.$$

I let you conclude. This will prove the derivability and it will gives you the value of $f'(x)$ for all $x$.

Surb
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