1

Complex Variables 2.2 Limits and Continuity

For real functions of real variables:

If we can find for a given positive we can obtain a positive that

We call f0 a limit for f(x) as x tends to x0:

(in general )

EXAMPLE:

EXAMPLE: Step function:

which is impossible to resolve

0

0

0 xx

if

fxf

0)(lim0

fxfxx

)()( 0000 xfforxff

221321

3)(lim;21)(1

xx

xfxxfx

0010

:0,1)(;0,0)(

00

xfandxfbutx

soxxuxxu

Function is continuous at a point x0 if: otherwise -> discontinuous

EXAMPLE: : discontinuous, but : continuous

EXAMPLE:

: discontinuous

FOR COMPLEX FUNCTIONS:

2

Complex Variables

)()(lim 00

xfxfxx

?

1

1lim

21

xx

0,20,2

0,10,sin

)(

xx

xxx

x

xf

4

1

11

1

1

1lim

221

xx

DEFINITION (Limit)

(deleted neighborhood)

3

Complex Variables

Now: can be just (or /n) we obtain z in the deleted neighborhood of i

For a real function: the limit can be approached from right or left

For a continuous function:

x0

x0+ x0-

f(x0-) f(x0+)

4

Complex Variables

For a complex function: there are infinitely many paths to approach!

If a limits exists: all the paths should lead to the same value of the function at z0

!!!0

5

Complex Variables

z =0 x

y

f(z)=i(y+1) i

z =0 x

y

f(z)=(x+1) 1

6

Complex Variables

11lim;01

lim 1

2

z

z zz

EXAMPLE

7

Complex Variables

z =0

arg z - ?

x

y

arg z = -

arg z =

8

Complex Variables

So:

Condition (b) is not satisfied at z = i.

But

The function IS continuous everywhere

iizfix

32lim

00

lim zfzfizx

i2

9

Complex Variables

(a) f(z) z; zz = z2 ; z2 + z +1; z2 - 2z +1= ( z-1)2 are all continuous

but z2 + z +1/ ( z-1)2 is not continuous at z = 1

(b) f(z) cos(z2); exp(z2);…continuous for all z. But not 1/z2.

(c,d)

all continuous and is also continuous and in R

1Regiondiskinsincos zyiyezf x

xezf ezf