A functional equation is an equation whose unknowns are functions.

$f(x+y) = f(x) + f(y),\quad f(x\cdot y) = f(x) \cdot f(y)$ are examples of such equations. The systematic study of these didn't begin before the 1960's, although various mathematicians have been studying them before, including Euler and Cauchy just to mention a few.

Functional equations appear many places, for example, the gamma function and Riemann's zeta function both satisfy functional equations.