Relating with fix point theorem and continuity

[Design]

Homework Statement

Assume the function f : [0,1] x [0,1] -> [0,1] is continuous and apply the IVT to prove that there is a number c E [0,1] such that f(c,y₀) = c for some y₀ E [0,1]

The Attempt at a Solution

I tried to break the cube up with the ranging being y₀ but I don't know how it maps y₀ to [0,1] to be continuous, if I can prove this then I can use the IVT.

thankyou

 

[micromass]

Let y₀ be fixed. Consider the function g(x)=f(x,y₀)-x. This function is sometimes \leq 0 and sometimes \geq 0. So you can apply the IVT on g.

 

[Design]

After you apply the IVT on g, you will get that the number between is zero?

 

[micromass]

Yes. You will get that there exists a c such that g(c)=0. This will be the c you're looking for.

 

[Design]

What do you mean when you mean y0 is fixed?

 

[micromass]

Just take y0 arbitrary.

 

[Design]

So y0 in the element of [0,1] right? and one side I got g(x) >= 0 >= g(x)-1, Is this right?

 

[micromass]

Yes!