MathDB

Let

H

be a set of real numbers that does not consist of

0

alone and is closed under addition. Further, let

f(x)

be a real-valued function defined on

H

and satisfying the following conditions:

\;f(x)\leq f(y)\ \mathrm{if} \;x \leq y

and f(x\plus{}y)\equal{}f(x)\plus{}f(y) \;(x,y \in H)\ . Prove that f(x)\equal{}cx on

H

, where

c

is a nonnegative number. [M. Hosszu, R. Borges]

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