A Note on Monomials

Abstract

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.

A dichotomy property for the graphs of monomials

“…It is known that f satisfies (1) if and only if f (x) = F (x, · · · , x) for a certain multi-additive and symmetric function F : R n → R, and that f is a polynomial function of degree at most n (i.e., f solves Fréchet's functional equation ∆ n+1 h f (x) = 0) if and only if f (x) = n k=0 f k (x), where f k (x) is a k-monomial function for k = 0, 1, · · · , n. (See, for example, [3], [6], for the proofs of these claims).…”

A qualitative description of graphs of discontinuous polynomial functions

“…where f : X → Y denotes a function, X, Y are two Q-vector spaces, and ∆ k h f (x) is defined inductively by ∆ 1 h f (x) = f (x + h) − f (x) and ∆ k+1 h f (x) = ∆ 1 h ∆ k h f (x), k = 1, 2, · · · . A simple induction argument shows that (1) can be explicitly written as (2) ∆ where f : R → R and ∆ h1h2···hs f (x) = ∆ h1 (∆ h2···hs f ) (x), s = 2, 3, · · · . Indeed, thanks to a classical result by Djoković [6], the equation with variable steps ∆ h1h2···hm+1 f (x) = 0 is equivalent to the equation with fixed step ∆ m+1 h f (x) = 0.…”

“…This result was firstly proved for the Cauchy functional equation by Kormes in 1926 [8]. Later on, in 1959, the result was proved for polynomials by Ciesielski [4] (see also [10], [11], [12], [15] [5], [14] for the original result, which was stated for solutions of the Cauchy functional equation and [1], [2], [15] for a direct proof of this result with polynomial functions).…”

“…Later on, in 1959, the result was proved for polynomials by Ciesielski [4] (see also [10], [11], [12], [15]). A weaker result is the so called Darboux type theorem, which claims that the polynomial function f : R → R is an ordinary polynomial if and only if f |(a,b) is bounded for some nonempty open interval (a, b) (see [5], [14] for the original result, which was stated for solutions of the Cauchy functional equation and [1], [2], [15] for a direct proof of this result with polynomial functions).…”

“…In [1], [2] Fréchet's equation was studied from a new fresh perspective. The main idea was to use the basic properties of Lagrange interpolation polynomials in one real variable.…”