How to find the derivative of function using given functional equation?

Question

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

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

$$f'(u)=?$$

Answer 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 $x$.