The functional equation f(y − x) − g(xy) = h (1/x− 1/y) is solved for general solution. The result is then applied to show that the three functional equations f(xy) = f(x)+f(y), f(y−x)−f(xy) = f(1/x−1/y) and f(y−x)−f(x)−f(y) = f(1/x−1/y) are equivalent. Finally, twice differentiable solution functions of the functional equation f(y − x) − g1(x) − g2(y) = h (1/x− 1/y) are determined.

We prove the generalized Hyers-Ulam-Rassias stability of a partitioned functional equation. It is applied to show the stability of algebra homomorphisms between Banach algebras associated with partitioned functional equations in Banach algebras.