0
$\begingroup$

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.

1 Answers 1

0

For $a

  • 0
    in this case, how did you know that induction on n would work? And also, what led you to think on doing the induction on a term of the form $x = s_0 + k \dot 2^{-n} \dot \frac{1}{t_0-s_0}$? I don’t think I could have come up with such an expression of $x$ to do induction. Thanks!2018-10-22