Brouwer fixed point theorem

I just learned the Brouwer Fixed Point Theorem which states that , the 2-sphere, has the fixed point property. I understand most part of the proof by contradiction by assuming that for any map , then for every , but I got stuck on why the function defined by where is the intersection of the ray starting at passing through and leaves , is a continuous function. The author (Hatcher) gives an explanation for it, but I can't really justify it with a rigorous proof. He said that "continuity of is clear since small pertubations of x produces small pertubations of , hence also small pertubations of the ray through these two points" How can I turn this into an proof?