Brouwer fixed point theorem

I just learned the Brouwer Fixed Point Theorem which states that $\displaystyle D^2$, the 2-sphere, has the fixed point property. I understand most part of the proof by contradiction by assuming that for any map $\displaystyle f: D^2 \rightarrow D^2$, then $\displaystyle f(x)\neq x$ for every $\displaystyle x\in D^2$, but I got stuck on why the function $\displaystyle r(x): D^2 \rightarrow S^1$ defined by $\displaystyle r(x)=p_x$ where $\displaystyle p_x$ is the intersection of the ray starting at $\displaystyle f(x)$ passing through $\displaystyle x$ and leaves $\displaystyle D^2$, 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 $\displaystyle r$ is clear since small pertubations of x produces small pertubations of $\displaystyle f(x)$, hence also small pertubations of the ray through these two points" How can I turn this into an $\displaystyle \epsilon-\delta$ proof?