If we replace by
Then it becomes
It implies that so
Is there another function satisifying the requirement ?
The equation that occurs in simplependulum's reply is known as Jensen's equation.
To solve it, put and then where .
Put in this latter equation: .
And now we see that .
Finally, put . Then . This is the well-known Cauchy's equation.
The most general continuous solution of Cauchy's equation is where is constant. Therefore is the most general continuous solution of Jensen's equation.
However, it must be said that there exist solutions of Cauchy's equation on which are not continuous, and they turn out to be extremely pathological. For example, if is one of these solutions and is any open interval then is dense in .
So it makes sense in many cases to ask for conditions on the solution, such as continuity, or boundedness on a finite interval, or monotonicity, anything which would avoid these weird functions.