# Simple Functional Equation

Find the function $f : \mathbb{R} \longrightarrow \mathbb{R}$ that for any $x, y \in \mathbb{R}$ satifies the following equation:

f(x + y) = f(x) - f(y)

# Substitution Method.

For $x = 0$ .

\begin{equation} f(y) = f(0) - f(y) \implies f(y) = \frac{1}{2}f(0) \end{equation}

For $x = y = 0$

\begin{equation} f(0) = f(0) - f(0) \implies f(0) = 0 \end{equation}

Combining equations $(1)$ and $(2)$ we can see that $f(y) = 0$ . This means that $f$ is a constant function equal to $0$ .

This method uses properties of addition.

For any two numbers $x, y \in \mathbb{R}$ .

\begin{align*} &f(x + y) = f(y + x) \\ &f(x) - f(y) = f(y) - f(x) \\ &2f(x) = 2f(y) \\ &f(x) = f(y) \\ &\implies f \text{ is constant} \\ &\implies f(x + y) = f(x) - f(y) = 0 \end{align*}

From that we see that for any number $k \in \mathbb{R}$ , $f(k) = 0$ .