Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R, C) to give a new proof of classical Montel’s theorem, about continuous solutions of Frechet’s functional equation ∆m h f = 0, for real functions (and complex functions) of one real variable. In this paper we use similar ideas to prove a Montel’s type theorem for the case of complex valued functions defined over the discrete group Z d. Furthermore, we also state and demonstrate an improved version of Montel’s Theorem for complex functions of several real variables and complex functions of several complex variables.

We study discontinuous solutions of the monomial equation 1 n! ∆ n h f (x) = f (h). In particular, we characterize the closure of their graph, G(f ) R 2 , and we use the properties of these functions to present a new proof of the Darboux type theorem for polynomials and of Hamel's theorem for additive functions.

Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R, C) to give a new proof of classical Montel's theorem, about continuous solutions of Fréchet's functional equation ∆ m h f = 0, for real functions (and complex functions) of one real variable. In this paper we use similar ideas to prove a Montel's type theorem for the case of complex valued functions defined over the discrete group Z d . Furthermore, we also state and demonstrate an improved version of Montel's Theorem for complex functions of several real variables and complex functions of several complex variables.

We prove that, if f : R n → R satisfies Fréchet's functional equationand f (x 1 , · · · , xn) is not an ordinary algebraic polynomial in the variables x 1 , · · · , xn, then f is unbounded on all non-empty open set U ⊆ R n . Furthermore, the set G(f ) R n+1 contains an unbounded open set.

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