Lecture6 : Properties of Continuous Functions[ Section 6.2 : Basic properties of Continuous Functions ]
(iii) i.e., maps closed bounded intervals to closed bounded intervals. We state next another important theorem.
6.2.8
Theorem (continuity of the inverse function) :
Let be continuous. If is one-one, then is either strictly increasing, i.e., whenever or it is strictly decreasing, that is, whenever . Further, if is the range of, then is one-one and continuous.
6.2.9
Theorem (Existence of the nth-root function) :
For every positive integer , the function is a one-one onto function. The inverse function, called the nth-root function, is also continuous, and is denoted by .
Practice Excercises :
1.
Show that a polynomial of odd degree has atleast one real root.
2.
Let be a continuous function such that and . Use theorem 6.2.7 suitably to show that there exist a point such that either for every .
3.
Show that the polynomial has at least two real roots.
4.
Let be a continuous function which takes only rational values. Show that is constant function.
5.
Give the examples of continuous functions with the following properties:
(i)
It maps a finite interval onto an infinite interval.
(ii)
It maps an infinite interval onto a finite interval.
i.e., 
maps closed bounded intervals to closed bounded intervals.
be continuous. If 
whenever 
or it is strictly decreasing, that is, 
whenever 
. Further, if 
is the range of
is one-one and continuous.
, the function
is a one-one onto function. The inverse function
, called the 
.
be a continuous function such that 
and 
. Use theorem 6.2.7 suitably to show that there exist a point 
such that either 
for every 
. 
has at least two real roots. 
