Properties of Continuous Function

Theorem :- 

A continuous function which has opposite signs at two points vanishes at least once between these points , that is if   f(x)   be continuous in the closed interval    [a,b]   and    f(a)    and    f(b)   have opposite signs , then there is at least one value of   x   between   a   and   b   for which    f(x)=0  .  

Proof :-

For the sake of definiteness , let us suppose that ;

F(a)<0   and    f(b)>0   .

Science   f(x)  is continuous and   f(a)<0  , therefore   f(x)   will be negative in the neighborhood of   a   .
Again Since    f(x)   is  continuous and     f(b)>0   therefore   f(x)   will be positive in the neighborhood of   b   .
The set of values of   x   between   a   and   b   which make   f(x)    positive is bounded below by  a  and hence possesses an exact lower bound    k  .

Hence ;   a<k<b 

In this way we find that   f(x)   is positive in the interval
k<x<b   and is negative or zero in the interval
a   is less then and equal to   x<k  .

Since   f(x)  is continuous at  x=k  , therefore by the definition of continuity,
f(k-0)=f(k)=f(k+0)  .

Since   f(x)   is negative or zero in
a   is less then and equal to    x<k ,   therefore    f(k-0)   must be negative and zero , therefore    f(k)    which is equal to    f(k-0)    must be negative or zero .

We shall now show that    f(k)   can not be negative  .

Since    f(x)    is positive in the interval   k<x<b   , that is ;
b>x>k   , therefore    f(k+0)   can not be negative and since ;
f(k)=f(k+0)   , therefore   f(k)   can not be negative

Hence it follows that    f(k)=0     and the Theorem is therefore proved .