I am thinking whether Every continuous function from $[0,1)$ on to $ [0,1)$ has a fixed point. I know about fixed point theorem for compact sets. The proof I came up fory mentioned problem is, if $f(0)=0$ then, we are done. If not, then it has to reach zero Somewhere in between and go to 1. So a compact subset of image is contained in the compact subset of domain or compact subset of domain is contained in the compact subset of image and thus it has a fixed point. Is it right??
Is it true that every interval mapped to itself with a continuous onto function, has a fixed point?