Module 2 : Limits and Continuity of  Functions
Lecture 6 : Properties of Continuous Functions [ Section 6.2 : Basic properties of Continuous Functions ]
6.2 
Some Basic Property of continuous functions
 

In this section we shall see how continuity of a function helps us to get a better picture of the function. We shall only state the results. For proofs, one may refer any book on Real Analysis.

6.2.1
Theorem  (Intermediate Value Property) :
 
Let be a continuous function. Let  be any real number between*and*. Then there exists some such that

Suppose and is a real number between*and *. Let us try to draw a picture of this.

Let be such that .  Saying that is continuous implies then its graph will cut the horizontal line
at least at one point .
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6.2.2
Corollary :
 
Let be a continuous function. Then, for every interval , the image set is also an interval. Thus, a continuous function maps intervals to intervals.
6.2.3
Example :
 
Let be defined by
                          
                                 

The function is not continuous at x =1. Observe that

                                   ,
1
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