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 real number plane.

Proof
From Additive Function is Linear for Rational Factors:
 * $\map f q = q \map f 1$

for all $q \in \Q$.

, let:
 * $\map f q = q$

for all $q \in \Q$.

Since $f$ is not linear, let $\alpha \in \R \setminus \Q$ be such that:
 * $\map f \alpha = \alpha + \delta$

for some $\delta \ne 0$.

Consider an arbitrary nonempty circle in the plane.

Let:
 * its centre be $\tuple {x, y}$ where $x \ne y$ and $x, y \in \Q$
 * its radius be $r > 0$.

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

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

Since $\delta \ne 0$, let $\beta = \dfrac {y - x} \delta$.

Since $x \ne y$:


 * $\beta \ne 0$

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


 * $\size {\beta - b} < \dfrac r {2 \size \delta}$

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


 * $\size {\alpha - a} < \dfrac r {2 \size b}$

Now put:


 * $X = x + b \paren {\alpha - a}$


 * $Y = \map f X$

Then:


 * $\size {X - x} = \size {b \paren {\alpha - a} } < \frac r 2$

so $X$ is in the circle.

Then:

Therefore
 * $\size {Y - y} = \size {b \paren {\alpha - a} - \delta \paren {\beta - b} } \le \size {b \paren {\alpha - a} } + \size {\delta \paren {\beta - b} } \le r$

so $Y$ is in the circle as well.

Hence the point $\tuple {X, Y}$ is inside the circle.

Also see

 * Definition:Additive Function
 * Definition:Cauchy Functional Equation
 * Definition:Everywhere Dense