Proof that a continuous function from [0,1] to [0,1] has a fixed point

Ananya0107 · Mar 25, 2015

Let f:[0,1]→[0,1] be a continuous function . Show that there exists a point such that f(x) = x

HallsofIvy · Mar 25, 2015

That's a fairly standard homework exercise. What have you tried?

Ananya0107 · Mar 25, 2015

Tell me if this argument is right... I constructed a 1by 1 square on the coordinate plane vertices (0,0) ,(0,1),(1,0),(1,1). Then I said that to graph any continuous function , the graph must touch or cut the square's diagonal , that is y=x.

AdityaDev · Mar 25, 2015

Apply MVT.
According to MVT, there exists a point c such that, f'(c)={f(1)-f(0)}/(1-0)=1-0=1
Now dy/dx=1. So you can find y.
now substitute a point in the domain to get the value of integration constant.

Proof that a continuous function from [0,1] to [0,1] has a fixed point

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Proof that a continuous function from [0,1] to [0,1] has a fixed point

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