Graph of Nonlinear Additive Function is Dense in the Plane

Theorem
Let $f: \R \to \R$ be an additive function which is not linear.

Then the graph of $f$ is dense in the plane.

Proof
From Additive Function is Linear for Rational Factors:


 * $f(q) = q f(1)$ for all $q\in\Q$.

Without loss of generality, let


 * $f(q) = q$ for all $q\in\Q$

and


 * $f(\alpha) = \alpha+\delta$ for some $\alpha\in\R\setminus\Q$ and $\delta \neq 0$.

Consider an arbitrary nonempty circle in the plane.

Let its centre be


 * $(x,y)$ where $x\neq y$ and $x,y\in\Q$

and its radius be $r>0$.

We will show how to find a point of the graph of $f$ inside this circle.

As $x\neq y$ and $r$ can be arbitrarily small, this will prove the theorem.

Since $\delta\neq0$, let


 * $\beta = \frac{y - x}{\delta}$

Since $x\neq y$,


 * $\beta\neq0$.

As Rationals are Everywhere Dense in Reals, there exists a rational number $b\neq 0$ such that:


 * $\left\vert \beta - b  \right\vert < \frac{r}{2 \left\vert\delta\right\vert}$

As Rationals are Everywhere Dense in Reals, there also exists a rational number $a$ such that:


 * $\left\vert \alpha - a  \right\vert < \frac{r}{2\left\vert b\right\vert} $

Now put:


 * $X = x + b (\alpha - a) \ $


 * $ Y = f(X) \ $

Then:


 * $|X-x| = |b (\alpha - a)| < \frac{r}{2}$

so $X$ is in the circle.

Then:

Therefore
 * $|Y-y| = |b (\alpha - a) - \delta (\beta - b)| \le |b (\alpha - a)| + |\delta (\beta - b)| \le r$

so $Y$ is in the circle as well.

Hence the point $(X, Y)$ is inside the circle.