Hint to continuity of retraction map for Standard Proof of Brouwer Fixed Point Theorem in Alg Top.

Question

I am looking for a hint to the continuity of the retraction map constructed in the usual manner for most intro algebraic topology courses (this is just self-study). I'm talking about $r:{D}\to \partial{D}$ that finds the intersection of the ray $[f(x),x,\infty)$ with the boundary of the circle. $f:D\to \partial D$ is continuous and so this seems intuitively true and I think of it as small perturbations in $x$ lead to small changes in $f$ and hence $r$ hardly changes either. Any hints? It really seems to come down to a basic understanding of geometry, of which I have a very weak background.

Answer 1

Suppose $\partial D$ is the unit sphere. The ray is $$f(x) + (x-f(x))t,\ t\ge 0.$$ $f(x) + (x-f(x))t\in\partial D$ is equivalent to $$1=\|f(x) + (x-f(x))t\|^2 = (f(x) + (x-f(x))t)\cdot(f(x) + (x-f(x))t) = \cdots$$ (where $\cdot$ is the scalar product)