# Continuity of Polynomials in the Complex Plane

Continuity of Polynomials in the Complex Plane

This Demonstration shows polynomials with zeros (or roots) of varying multiplicity in the complex plane. The object is to show that such a polynomial is a continuous function at a selected point .

z

0

Let be a complex-valued function, where . The function is continuous at a point if for every there is a such that for all points that satisfy the inequality , the inequality holds.

f:EC

E⊂C

f(z)

z

0

ϵ>0

δ>0

z∈E

|z-|<δ

z

0

f(z)-f()<ϵ

z

0

Assign a color to each point of the complex plane as a function of , namely the RGB color of three arguments , , (red, green, blue). If (with chosen by slider), use black. Otherwise, if , let ; if , let ; if , let .

z

w=f(z)-f()

z

0

r

g

b

|w|<ϵ

ϵ

Re(w)>0

r=1

Im(w)Re(w)<0

g=1

Re(w)<0

b=1

The zeros of the polynomial can be set using the three locators, and their respective multiplicities can be selected. The movable white marks the point .

×

z

0

A black patch around means that . Subsequently, finding a such that the circle is inside the patch verifies continuity.

z

0

w<ϵ

δ

z-<δ

z

0

Use high resolution after setting all the arguments and parameters.