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 $$\begin{equation} \forall \, x, y \in \mathbb{R}^D \left( |x| = |y| \quad \implies \quad f(x) = f(y) \right) \end{equation}$$