Lecture6 : Properties of Continuous Functions [ Section 6.2 : Basic properties of Continuous Functions ]
however there is no value for which . This shows that the continuity condition in theorem 6.2.1 cannot be removed.
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 .
6.2.5
Example (locating zeros) :
Consider the function .
It is not easy to find solutions of the equation . However, we observe that
andis a continuous function. Hence by theorem 2.5.1, there exists such that . Thus we are able to locate a zero of in the interval (-1, 0).
6.2.6
Example :
(i)
Let be defined by . Then, is continuous, but is not bounded on say .
(ii)
Let be defined by . Then is continuous, is bounded but does not attain a maximum value or a minimum value on.
6.2.7
Theorem (maximum and minimum) :
Let be continuous. Then the following hold:
(i) is bounded.
(ii)attains its maximum and minimum, that is, there exist and in such that , for every and , for every . Further range of where .
for which 
be a continuous function. Then there exists some 
such that 
. Such a point is called a 
. 
.
. However, we observe that
is a continuous function. Hence by theorem 2.5.1, there exists 
such that 
. Thus we are able to locate a zero of
in the interval 
be defined by 
. Then, 
is continuous, but 
. 
be defined by 
. Then 
is continuous, 
is bounded but 
does not attain a maximum value or a minimum value on
be continuous. Then the following hold:
is bounded.
and 
in 
such that 
, for every
and 
, for every 
. Further range
where 
.
