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Jensen with weaker condition?

Source: VJIMC 1998 2.4-M

August 24, 2021

inequalitiesFunctional Analysisfunctionreal analysis

Problem Statement

A function

f:\mathbb R\to\mathbb R

has the property that for every

x,y\in\mathbb R

there exists a real number

t

(depending on

x

and

y

) such that

0<t<1

and

f(tx+(1-t)y)=tf(x)+(1-t)f(y).

Does it imply that

f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2

for every

x,y\in\mathbb R

?

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