TITLE: Finding the value of $f(2)$ if $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$, $f(0)=1$ and $f'(0)=-1$
QUESTION [5 upvotes]: Let $f:\mathbb{R}\to\mathbb{R}$ be a function satisfying the conditions
$$\begin{gather}
f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} \tag{1}\\
f(0)=1 \tag{2}\\
f'(0)=-1 \tag{3}
\end{gather}$$
Find the value of $f(2)$ by proper explanation.

REPLY [4 votes]: From (1) and (2): $2^k\left(f(2^{1-k})-1\right)=f(2)-1$ for all $k \geq 0$.
From (3): $\lim_{k\to \infty}2^k\left(f(2^{1-k})-1\right)=-2$.
So $f(2)=-1$.