Difficulty with Jensen's Equation.

Question

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$

But I am finding difficulty in finding all function continuous on $(a,b),a,b \in \Bbb R$, satisfying the Jensen equation.

Some related posts: How to show that $f$ is a straight line if $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$? and Function that is both midpoint convex and concave. — Martin Sleziak, Aug 08 '17 at 04:36

score 0 · Answer 1 · answered Jan 25 '15 at 16:21

0

Hint: First solve the question for functions defined on a closed interval, rather than open. Then think of an open interval as the (infinite) union of closed intervals.

answered Jan 25 '15 at 16:21

Amitai Yuval

19,308

Difficulty with Jensen's Equation.

1 Answers1