Monotonic additive function. I need to show it's linear

Question

If a real additive function f is monotonic, then it is linear.

I need to show that the monotonic function f that satisfies caushy additives functional equation is linear

Answer 1

Here is the outline of a proof:

Using the fact that $f$ is additive, show that

• $f(0)=0$
• $f(x)=x\cdot f(1)$ for $x=0,1,2,\ldots$
• $f(x)=x\cdot f(1)$ for $x\in \mathbb Z$
• $f(x)=x\cdot f(1)$ for $x\in \mathbb Q$

Then use the fact that $f$ is monotonic to show that $\lim_{x\to a}f(x)=f(a)$ for all real $a$.

Then show that $f(x)=x\cdot f(1)$ for $x\in \mathbb R$.

That shows that $f$ is linear.

There are variations on this proof, of course. Let us know if you need more details.