# The Mean Value Property and Affine Functions

I need some hints to solve the following:

A function $f(t)$ on an interval $I = (a,b)$ has the mean value property if $f(\frac{s+t}{2}) = \frac{f(s)+f(t)}{2}$ where $s,t\in{I}$. Show that any continuous function on $I$ with the mean value property is affine.

This problem is taken from Chapter 3, Section 4, Question 3 from Theodore Gamelin's Complex Analysis.

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For $a<s_0<t_0<b$ let $m=\frac{f(t_0)-f(s_0)}{t_0-s_0}$ and $b=f(s_0)-ms_0$. Thus we see that $$\tag1f(x) = mx+b$$ holds at least for $x\in\{s_0,t_0\}$. Show by induction on $n$ that $(1)$ holds for all $x$ of the form $x=s_0+k\cdot 2^{-n}\cdot \frac 1{t_0-s_0}$ with $0\le k\le 2^n$. Then conclude by continuity that $(1)$ holds for all $x\in[s_0,t_0]$. If you selected $s_0,t_0$ close enough to $a$ and $b$ respectively, you have $2x-s_0\in[s_0,t_0]$ for all $x \in(a.s_0)$ and can use that and $f(x)=2f(s_0)-f(2x-s_0)$ to conclude that holds for $t\in(a,s_0)$ as well. Treat $x\in(t_0,b)$ similarly.

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