Hi,

I would like to show that a continuous map $\displaystyle f

a,b)\to \mathbb{R}$ which has the property that $\displaystyle f\left(\frac{x+y}{2}\right) = \frac{f(x)+f(y)}{2}$ for $\displaystyle x,y\in (a,b)$ satisfies

$\displaystyle f(x) = Ax + B$ for some constants $\displaystyle A, B$.

I can prove it on the closed interval [0,1]: in this case it suffices to show $\displaystyle f(x)=Ax$ with $\displaystyle f(0)=0$. I prove it holds on the dyadic rationals of the form $\displaystyle \frac{m}{2^k}$ by an induction argument, and extending to the closed interval by continuity.

**BUT** my argument needs the endpoints of the interval: I am progressively subdividing; $\displaystyle f(\frac{1}{2}) = \frac{f(0)+f(1)}{2} = \frac{f(1)}{2}$, etc, and I cannot see how to prove it on an

**open** interval.

Many thanks