TITLE: If $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$, then find $f(2)$
QUESTION [8 upvotes]: Let $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ and it is given that 
$f(0)=1$ and $f'(0)=-1$, where $f'$ denotes first derivative. Find the value of $f(2)$
Could someone tell me how to use $f'(0)=-1$ here? I am not able to use this information.

REPLY [8 votes]: If $f$ satisfies your equation, then $g(x) = f(x) - f(0)$ satisfies the Cauchy functional equation $g(x+y) = g(x) + g(y)$.  Of course, if $f'(0)$ exists, then $g$ is continuous.  The only continuous solutions of the Cauchy functional equation are the linear functions $g(x) = ax$, so the only continuous solutions of your equation are the affine functions $f(x) = a x + b$.  The rest is easy.