# Continuity of a Complex Function

Continuity of a Complex Function

Let be a complex function where and are open subsets in . The function is continuous at the point if for every there is a such that for all points that satisfy the inequality , the inequality holds.

f:EF

E

F

f(z)

z

0

ϵ>0

δ>0

z∈E

|z-|<δ

z

0

f(z)-f()<ϵ

z

0

We assign a color to each point of the complex plane as a function of , namely the RGB color of four arguments , , , and (red, green, blue, and opacity). If (with chosen by the slider), we use black. Otherwise, if , ; if , ; if , .

z

w=f(z)-f()

z

0

r

g

b

o

|w|<ϵ

ϵ

Re(w)>0

r=1

Im(w)Re(w)<0

g=1

Re(w)<0

b=1

A black patch around the point means that the function has . After that, we find a such that the circle is inside the patch. Note the branch cut for along the negative real axis.

z

0

w<ϵ

δ

z-<δ

z

0

z