Theorem (First Fixed Point Theorem). Assume that , i. e.    is continuous on  .  
Then we have the following conclusions.
(i).    If the range of the mapping satisfies for all , then   has a fixed point in .
(ii).    Furthermore, suppose that is defined over and that a positive constant exists with
      for all  ,  then has a unique fixed point in .    

Proof of (i).

If    or  ,  then the assertion is true.  Otherwise, the values of and must satisfy  

     (a,b]   and    [a,b).

The function   has the property that

       and      .  
  
Now apply the Intermediate Value Theorem to   with the constant  ,  

and conclude that there exists a number  ,  with   for which   .   

It follows that,   and    is the desired fixed point of  ,  which establishes (i).  

Q. E. D.

(c) John H. Mathews 2004