Example 2.18.  Show that the polynomial function given by  

              

is continuous at each point   in the complex plane.

Solution.  If    is the constant function, then  ;  and if  ,  then we can use Definition 2.3 with    and the choice    to prove that    .  Using Property (2-19) and mathematical induction, we obtain  

(2-27)            ,   for   .  

We can extend Property (2-18) to a finite sum of terms and use the result of Equation (2-27) to get   

            .  

Conditions (2-24),  (2-25), and (2-26) are satisfied, so we conclude that P is continuous at .

Explore Solution 2.18.

For illustration, we use  n = 5.
Enter the function P[z].

For illustration, we use  n = 105.
Enter the function P[z].

Therefore  P(z)  is a continuous function for all complex numbers.

(c) 2006 John H. Mathews, Russell W. Howell