For real numbers , and > 0 , the inequality | - | means that
- +,
i.e., [ -, + ] .
1.2.8
Definitions (Algebra of real valued functions) :
Let and.
(i) The sum of and g is the function + g :, defined by ( + g ) ( ): = f ( ) + g ( ) , .
(ii) The product of and g is the function g : , defined by
( g ) ( ) : = ( ) g ( ), .
(iii) If g ( ) 0 , the quotient of and g is the function / g : , defined by
( / g ) ( ) : = ( ) / g ( ) , .
(iv) For any real number , the function is the function:, defined by
( ) ( ) : = ( ), .
(v) For any integer and real numbers , the function defined by ,
is called a polynomial function. When every is an integer, it is called an integral polynomial function.
, (
. 
and 
and g is the function 
0 
and real numbers
, the function 
defined by
,
is an integer, it is called an integral polynomial function. 
