WOLFRAM|DEMONSTRATIONS PROJECT

Continuity of a Complex Function

​
z
2
z
z
3
z
sin(z)
cos(z)
tan(z)
cot(z)
z

log(z)
sinh(z)
cosh(z)
tanh(z)
coth(z)
Axes
plot range
0.1
0.2
0.5
1.
1.5708
3.14159
6.28319
ϵ
1
δ
0.1
Let
f:EF
be a complex function where
E
and
F
are open subsets in

. The function
f(z)
is continuous at the point
z
0
if for every
ϵ>0
there is a
δ>0
such that for all points
z∈E
that satisfy the inequality
|z-
z
0
|<δ
, the inequality
f(z)-f(
z
0
)<ϵ
holds.
We assign a color to each point
z
of the complex plane as a function of
w=f(z)-f(
z
0
)
, namely the RGB color of four arguments
r
,
g
,
b
, and
o
(red, green, blue, and opacity). If
|w|<ϵ
(with
ϵ
chosen by the slider), we use black. Otherwise, if
Re(w)>0
,
r=1
; if
Im(w)Re(w)<0
,
g=1
; if
Re(w)<0
,
b=1
.
A black patch around the point
z
0
means that the function has
w<ϵ
. After that, we find a
δ
such that the circle
z-
z
0
<δ
is inside the patch. Note the branch cut for
z
along the negative real axis.