LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-08-06 (claude-sonnet-5)

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Radial normalisation $x \mapsto x / ∥ x ∥_{2}$ is continuous on $R^{n} ∖ {0}$

Statement

For $n \geq 1$ , the map $ρ : R^{n} ∖ {0} \to S^{n - 1}$ defined by $ρ (x) = x / ∥ x ∥_{2}$ is continuous.

Facts & Assumptions

Given: $n \geq 1$ , the Euclidean norm, and a nonzero point $a \in R^{n}$ .

[L1]

The Euclidean norm is continuous and satisfies $∣ ∥ u ∥_{2} - ∥ v ∥_{2} ∣ \leq ∥ u - v ∥_{2}$ (The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ ).

[L2]

Componentwise continuity gives continuity into $R^{n}$ (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, Vector-valued functions $f : A \to R^{m}$ , their limits and continuity, with the dictionary to the metric notions); and a map into a subspace is continuous exactly when its composite with the ambient inclusion is continuous, so a continuous map whose image lies in the subspace is continuous into it (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

[L3]

The unit sphere is the set of vectors with Euclidean norm $1$ (Euclidean spheres and closed balls as subspaces of $R^{n}$ ).

Proof

technique · direct

1.1L1

Put $d : = ∥ a ∥_{2} > 0$ . If $∥ x - a ∥_{2} < d / 2$ , then [L1] gives $∥ x ∥_{2} > d / 2$ .

1.2L1

For such $x$ , $∥ ρ (x) - ρ (a) ∥_{2} \leq ∥ x - a ∥_{2} / ∥ x ∥_{2} + ∥ a ∥_{2} ∣ 1 / ∥ x ∥_{2} - 1 / ∥ a ∥_{2} ∣ \leq 4 ∥ x - a ∥_{2} / d$ .

2.1L2L3step 1.2∎

Step 1.2 gives the epsilon-delta condition at $a$ , so $ρ$ is continuous on the punctured space. Also $∥ ρ (x) ∥_{2} = 1$ , so its image lies in $S^{n - 1}$ and [L2] gives continuity with that codomain.

Radial normalisation $x \mapsto x / ∥ x ∥_{2}$ is continuous on $R^{n} ∖ {0}$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Radial normalisation x↦x/∥x∥2x↦x/∥x∥2​ is continuous on Rn∖{0}Rn∖{0}

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Radial normalisation $x \mapsto x / ∥ x ∥_{2}$ is continuous on $R^{n} ∖ {0}$