# Thread: Brouwer fixed point theorem

1. ## Brouwer fixed point theorem

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

2. Take $1=|(1-s)x+sf(x)|$, solve for $s\in \mathbb{R}$ and pick the nonnegative root ( $|.|$ denotes euclidean distance).

3. Also, you only need to prove that the map which takes a point of the disc to its corresponding circle point is cont. since your map is the composition of that with $f$. But, that map, call it $R$, is given by $R:ce^{i\theta}\mapsto e^{i\theta}$ where $ce^{i\theta}$ is the unique representation of an element of $\mathbb{D}^2$ as the product of something in $c\in(0,1)$ and $e^{i\theta}\in\mathbb{S}^1$. From there it's clear how to proceed.

4. Originally Posted by Jose27
Take $1=|(1-s)x+sf(x)|$, solve for $s\in \mathbb{R}$ and pick the nonnegative root ( $|.|$ denotes euclidean distance).
Thank you very much for your help, Jose, but I don't really follow the hint (Sorry. I'm slow on this). So you write the equation of the ray passing through $x$ and $f(x)$, and set its Euclidean distance equals to 1. So, I have $1=\mid s(f(x)-x)+x\mid$. I tempted to say that $s=\frac{1-x}{f(x)-x}$, but that does not seem right since the expression is Euclidean distance.

Thanks a lot for your help too, Drexel. I'll try to work on your hint tomorrow morning. My brain is not working now.