For $n\ge 1$, the map $H(x,t)=((1-t)+t\mathrm{/}\parallel x{\parallel }_2)x$ is continuous on $({\mathbb{R}}^n\setminus \{0\})\times [0,1]$, starts at $x$, ends at radial normalisation, fixes the unit sphere, and never reaches $0$

Given: $x\in P$ and $t\in [0,1]$.

Radial normalisation is continuous on $P$ (Radial normalisation $x\mapsto x\mathrm{/}\parallel x{\parallel }_2$ is continuous on ${\mathbb{R}}^n\setminus \{0\}$).

Coordinate projections and the map into a product are continuous as stated by the product universal property (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

The Euclidean norm is continuous for the Euclidean metric and is positive away from $0$, and the unit sphere consists of its norm-one points (The finite and reverse triangle inequalities for a norm; and for $n\ge 1$ every norm $N$ on ${\mathbb{R}}^n$ satisfies $N(x)\le C\parallel x{\parallel }_1$ and is Lipschitz, hence continuous, for $d_2$, Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$).

For maps on a subset of a metric space, sums, scalar multiples and pointwise products of continuous real-valued functions are continuous, and a vector-valued map is continuous exactly when its coordinate functions are continuous (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, clauses 1 and 3, the product being the case $m=1$ of the inner product); composites of continuous maps are continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous); and $u\mapsto 1\mathrm{/}u$ is continuous on $\mathbb{R}\setminus \{0\}$, this being clause 4 of (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function) with $A=\mathbb{R}$, numerator $1$ and denominator the identity.

A map into a subspace is continuous exactly when its composite with the ambient inclusion is continuous (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).