Functional Equations

A functional equation is an equation in which the function itself is the unknown, and the equation involves the function evaluated at various arguments. The goal is to determine all functions (often from $ext{R} o ext{R}$ or $ext{N} o ext{N}$) that satisfy the given relation for all inputs in the domain.

Important properties

• Functional equations often require finding all functions that satisfy a given property for every input.

• Common strategies include substituting specific values, guessing and checking, and using properties like symmetry.

• Some functional equations have unique solutions; others have infinitely many.

• Recursion is a special case where the value at one point depends on previous values.