1. ## fixed point mapping

Trying to figure out these two questions, got the rest of them but can't remember my linear algebra for the life of me thanks!

a. make a continuous function (0,1] --> (0,1] without fixed points.
note: there is supposed to be some simple map but honestly I can't see it.

b. prove that the unit square [0,1]x[0,1] is homeomorphic to the parellelogram in Rē with vertices (3,2), (6,5), (4,4), and (7,7). Use a suitable map and an appropriate translation.

Thanks so much for the help appreciated.

2. Originally Posted by ml692787
Trying to figure out these two questions, got the rest of them but can't remember my linear algebra for the life of me thanks!

a. make a continuous function (0,1] --> (0,1] without fixed points.
note: there is supposed to be some simple map but honestly I can't see it.
If it was [0,1] --> [0,1] it would be impossible by the the Brouwer fixed point theorem. However, here it is (0,1] hence take $\displaystyle f(x) = x^2$.

3. Originally Posted by ThePerfectHacker
However, here it is (0,1] hence take $\displaystyle f(x) = x^2$.
Surely $\displaystyle x=1$ is a fixed point for that mapping.

However, the following continuous function has no fixed point.
$\displaystyle f(x) = \left\{ {\begin{array}{cr} {x^2 } & {0 < x \le \frac{1}{2}} \\ {\left( {1 - x} \right)^2 } & {\frac{1}{2} < x \le 1} \\ \end{array}} \right.$

4. thanks for the help, i'm trying to figure out how to do the second part, I forgot how to map a vector <2,1> to the unit vector <0,1>, and the vector <3,3> to <1,0>, thanks