On each line $l \subset \mathbb{R}^n, n \geqslant 2$ passing through $0$ we choose a point $a(l)$ such that $a(l)$ depends continuously on $l$. I have to prove that there exists a line with $a(l)=0$.
I show it using Borsuk-Ulam theorem. We can consider the set of lines passing through the origin in $\mathbb{R}^n$ as sphere $\mathbb{S}^{n-1}$ (I fix this ambiguity later). We assign to each point on a sphere a real number $\lambda$ (point on a line passing through the origin and $x$ is $\lambda x$ for some $\lambda$). So we have a continuous function $$\varphi:\mathbb{S}^{n-1} \to \mathbb{R}$$ such that $\varphi(-x) = -\varphi (x)$, so it is odd. I take the embedding of $\mathbb{R}$ into $\mathbb{R}^{n-1}$ given by $x \longmapsto (x,0,\dots,0)$ which is continuous and this gives me a continuous odd function from $\mathbb{S}^{n-1} \to \mathbb{R}^{n-1}$. Finally, Borsuk-Ulam claims that there should be a point on a sphere with $\varphi(x)=\varphi(-x)$ and clearly, by oddity $\varphi(x)=0$ for some $x$. Am I correct? Thanks in advance.