Basic sets for some basic functional equations

Abstract Motivated by the well known fact that any nonzero solution of fundamental Cauchy functional equation may arbitrarily be prescribed on a Hamel basis, we deal with following problem: given $$\begin{aligned} E_1(\varphi ) = E_2(\varphi \end{aligned}$$ E 1 ( φ ) = 2 unknown function $$\varphi : X \longrightarrow Y$$ : X ⟶ Y , what must set $$\emptyset \ne Z \subset X$$ ∅ ≠ Z ⊂ like in order to ensure an arbitrary _0: 0 admits unique solving ( $$*$$ ∗ and such _{|_Z} \varphi _0$$ | ; if does exist it is termed basic set. We discuss problem existence sets for exponential functions, d’Alembert’s sine Cuculière’s functions hyperbolic tangent type among others.