Functional equation

Definition

Let $X$ and $Y$ be sets, let $k \ge 1$, and let $D \subseteq X^k$. Let $f \colon X \to Y$ be an unknown function. Let $E_1(f)$ and $E_2(f)$ be expressions built from $f$, from given functions and from variables $x_1, \dots, x_k$ by finitely many substitutions and compositions. A functional equation in $f$ is a condition $$E_1(f)(x_1, \dots, x_k) = E_2(f)(x_1, \dots, x_k) \quad \text{for all } (x_1, \dots, x_k) \in D.$$ In particular the expressions contain no derivatives, integrals or limits of $f$. A solution is a function $f \colon X \to Y$ that satisfies the condition. Systems of such conditions and several unknown functions are allowed in the same way.