# Problem 3.

Find all functions $f(x)$ defined on the entire number line and satisfying the condition

$$
f(x-y)=f(x) \cdot f(y) \text { for all } x, y.
$$

#
# Solution.

One solution is obvious, which is $f(x) \equiv 0$.

Consider functions that are not identically zero over the entire number line.

By setting $t=x-y$, we get that $f(t)=f(t+y) f(y)$. Therefore, $f(y)$ does not equal 0 for any $y$, because otherwise the value of $f(t)$ would be 0 for all $t$, which contradicts the problem's condition.

By setting $y=x / 2$, we obtain from the original equation that $f(x / 2)=f(x) f(x / 2)$, and we already know that $f(x / 2) \neq 0$. Dividing both sides by $f(x / 2)$, we get $f(x)=1$. Note that the number $x$ could be chosen arbitrarily, which completes the proof.

Answer: $f(x) \equiv 0, f(x) \equiv 1$, there are no others.