# Thread: Proof using the Intermediate Value Theorem

1. ## Proof using the Intermediate Value Theorem

A fixed point of a function f is a number c in its domain such that f(c) = c. (the function doesn't move the input value c; it stays fixed.) Suppose that f is a continuous function with domain [0,1] and range contained in [0,1]. Prove that f must have at least one fixed point.

2. Originally Posted by gmencl
A fixed point of a function f is a number c in its domain such that f(c) = c. (the function doesn't move the input value c; it stays fixed.) Suppose that f is a continuous function with domain [0,1] and range contained in [0,1]. Prove that f must have at least one fixed point.
Wow, what a beautify theorem. I never really appreciated fixed points theorems until now.

Define a function on the closed interval [0,1]
g(x)=f(x)-x
We note that since f(x) and x are continous on [0,1] so too g(x) is continous on [0,1].
We note that g(0)=f(0)-0=f(0)
And g(1)=f(1)-1
Here is the crucial steps:
f(0)=>0 by the defintion of f(x).
If f(0)=0 then the proof is complete, that is c=0.
Thus, there is not harm in assuming f(0)>0
f(1)<=1 thus, f(1)-1<=0
If f(1)-1=0 the proof is complete, that is c=1.
Thus, there is no harm in assuming f(1)-1<0
Since g(x) is continous by intermediate value theorem, [zero is between the f(0) and f(1)-1]
There is a point c in [0,1] such that, g(c)=0
But, g(c)=f(c)-c thus, f(c)=c.
Q.E.D.