ABSTRACT
The goal of this chapter is to examine the stability of the d’Alembert functional equation (also known as the cosine functional equation)
f(x+ y) + f(x− y) = 2 f(x) f(y) for all x, y ∈ R (20.1) and the sine functional equation
f(x+ y) f(x− y) = f(x)2 − f(y)2 for all x, y ∈ R. (20.2) In this chapter, we will show that if the function
(x, y) 7→ f(x+ y) + f(x− y)− 2f(x)f(y) is bounded on R2, then either f is bounded on R or f is a solution of the d’Alembert functional equation (20.1). This result was established for the first time by Baker (1980). In this chapter, we will also prove that if the function
(x, y) 7→ f(x+ y)f(x− y)− f(x)2 + f(y)2
is bounded on R2, then either f is bounded on R or f is a solution of the sine functional equation (20.2). Cholewa (1983) was the first mathematician to investigate the stability of sine functional equation.
