Intermediate value theorem problem

[shrug]

1. How can you prove if F is continuous, then there exists a fixed point of F in [0,1]?



I know F:[0,1] ---> [0,1] an bijective, but what is f(c)=c mean?

 

[morphism]

Have you tried drawing a picture? It might be enlightening.

 

[Dick]

I think it means more or less exactly what it says. There is a c in [0,1] such that F(c)=c. Use the intermediate value theorem. If F is bijective there is an a such that F(a)=0 and a b such that F(b)=1. What happens in between? Apply the IVT to F(x)-x on the interval [a,b].

 

[shrug]

I think it means more or less exactly what it says. There is a c in [0,1] such that F(c)=c. Use the intermediate value theorem. If F is bijective there is an a such that F(a)=0 and a b such that F(b)=1. What happens in between? Apply the IVT to F(x)-x on the interval [a,b].

Does that mean I need to pick a point like .5, which is between [0,1].

Also, would there be any point where F isn't continuous and a fixed point may not exist.

Thanks guys

 

[Dick]

I don't think you really understood what I wrote. As morphism said, draw a picture. Look up the IVT.

 

[HallsofIvy]

No, you can't "pick" a point. And you are told that the function is continuous. Why are you asking if it isn't?

If F is from [0, 1] what can you say about F(0)? What can you say about F(1)?

What can you say about 0- F(0) and 1- F(1)?