▾ Set of symbols
▾ Alphabet
▾ Deduction system
▾ Theory
▾ Zermelo-Fraenkel set theory
▾ Set
▾ Binary cartesian set product
▾ Binary relation
▾ Map
▾ Constant map
▾ Constancy-preserving map
▾ Radial map
Radial function
Formulation 1
Let $\mathbb{R}^D$ be a D816: Euclidean real Cartesian product such that
(i) $|\cdot|$ is the D1383: Euclidean length function on $\mathbb{R}^D$
(ii) $f : \mathbb{R}^D \to \mathbb{R}$ is a D4364: Real function on $\mathbb{R}^D$
Then $f$ is a radial function if and only if \begin{equation} \forall \, x, y \in \mathbb{R}^D \left( |x| = |y| \quad \implies \quad f(x) = f(y) \right) \end{equation}