Module 2 : Limits and Continuity of  Functions
Lecture 6 : Properties of Continuous Functions [ Section 6.2 : Basic properties of Continuous Functions ]
6.2.4
Corollary (Fixed point theorem) :
 
Let be a continuous function. Then there exists some such that . Such a point is called a fixed point for .                                                                                                   
   
 
Proof:
   
 

Consider the function .

Then, since  ,

Thus, by the theorem 6.2.3, there exists such that  , i.e. .                        Back
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