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By way of contradiction, suppose B s / 2 ( f ( c ) is not contained in f ( U ) , , and let w B s / 2 ( f ( c ) ) be a complex number that is not in f ( U ) . We have that | w - f ( c ) | < s / 2 , which implies that | w - f ( z ) | > s / 2 for all z C r . Consider the function g defined on the closed disk B ¯ r ( c ) by g ( z ) = 1 / ( w - f ( z ) ) . Then g is continuous on the closed disk B ¯ r ( c ) and differentiable on B r ( c ) . Moreover, g is not a constant function, for if it were, f would also be a constant function on B r ( c ) and therefore, by the Identity Theorem, constant on all of U , whichg is not the case by hypothesis. Hence, by the Maximum Modulus Principle,the maximum value of | g | only occurs on the boundary C r of this disk. That is, there exists a point z ' C r such that | g ( z ) | < | g ( z ' ) | for all z B r ( c ) . But then

2 s = 1 s / 2 < 1 | w - f ( c ) | < 1 | w - f ( z ' ) | 1 s ,

which gives the desired contradiction. Therefore, the entire disk B s / 2 ( f ( c ) ) belongs to f ( U ) , and hence the point f ( c ) belongs to the interior of the set f ( U ) . Since this holds for any point c U , it follows that f ( U ) is open, as desired.

Now we can give the version of the Inverse Function Theorem for complex variables.

Let S be a piecewise smooth geometric set, and suppose f : S C is continuously differentiable at a point c = a + b i , and assume that f ' ( c ) 0 . Then:

  1. There exists an r > 0 , such that B ¯ r ( c ) S , for which f is one-to-one on B ¯ r ( c ) .
  2.   f ( c ) belongs to the interior of f ( S ) .
  3. If g denotes the restriction of the function f to B r ( c ) , then g is one-to-one, g - 1 is differentiable at the point f ( c ) , and g - 1 ' ( f ( c ) = 1 / f ' ( c ) .

Arguing by contradiction, suppose that f is not one-to-one on any disk B ¯ r ( c ) . Then, for each natural number n , there must exist two points z n = x n + i y n and z n ' = x n ' + i y n ' such that | z n - c | < 1 / n , | z n ' - c | < 1 / n , and f ( z n ) = f ( z n ' ) . If we write f = u + i v , then we would have that u ( x n , y n ) - u ( x n ' , y n ' ) = 0 for all n . So, by part (c) of [link] , there must exist for each n a point ( x ^ n , y ^ n ) , such that ( x ^ n , y ^ n ) is on the line segment joining z n and z n ' , and for which

0 = u ( x n , y n ) - u ( x n ' , y n ' ) = t i a l u t i a l x ( x ^ n , y ^ n ) ( x n - x n ' ) + t i a l u t i a l y ( x ^ n , y ^ n ) ( y n - y n ' ) .

Similarly, applying the same kind of reasoning to v , there must exist points ( x ˜ n , y ˜ n ) on the segment joining z n to z n ' such that

0 = t i a l v t i a l x ( x ˜ n , y ˜ n ) ( x n - x n ' ) + t i a l v t i a l y ( x ˜ n , y ˜ n ) ( y n - y n ' ) .

If we define vectors U n and V n by

U n = ( t i a l u t i a l x ( x ^ n , y ^ n ) , t i a l u t i a l y ( x ^ n , y ^ n ) )

and

V n = ( t i a l v t i a l x ( x ˜ n , y ˜ n ) , t i a l v t i a l y ( x ˜ n , y ˜ n ) ) ,

then we have that both U n and V n are perpendicular to the nonzero vector ( ( x n - x n ' ) , ( y n - y n ' ) ) . Therefore, U n and V n are linearly dependent, whence

det ( ( tial u tial x ( ( x ^ n , y ^ n ) tial u tial y ( x ^ n , y ^ n ) tial v tial x ( x ˜ n , y ˜ n ) tial v tial y ( x ˜ n , y ˜ n ) ) ) = 0 .

Now, since both { x ^ n + i y ^ n } and { x ˜ n + i y ˜ n } converge to the point c = a + i b , and the partial derivatives of u and v are continuous at c , we deduce that

det ( ( tial u tial x ( a , b ) tial u tial y ( a , b ) tial v tial x ( a , b ) tial v tial y ( a , b ) ) ) = 0 .

Now, from [link] , this would imply that f ' ( c ) = 0 , and this is a contradiction. Hence, there must exist an r > 0 for which f is one-to-one on B ¯ r ( c ) , and this proves part (1).

Because f is one-to-one on B r ( c ) , f is obviously not a constant function. So, by the Open Mapping Theorem, the point f ( c ) belongs to the interior of the range of f , and this proves part (2).

Now write g for the restriction of f to the disk B r ( c ) . Then g is one-to-one. According to part (2) of [link] , we can prove that g - 1 is differentiable at f ( c ) by showing that

lim z f ( c ) g - 1 ( z ) - g - 1 ( f ( c ) ) z - f ( c ) = 1 f ' ( c ) .

That is, we need to show that, given an ϵ > 0 , there exists a δ > 0 such that if 0 < | z - f ( c ) | < δ then

| g - 1 ( z ) - g - 1 ( f ( c ) ) z - f ( c ) - 1 f ' ( c ) | < ϵ .

First of all, because the function 1 / w is continuous at the point f ' ( c ) , there exists an ϵ ' > 0 such that if | w - f ' ( c ) | < ϵ ' , then

| 1 w - 1 f ' ( c ) | < ϵ .

Next, because f is differentiable at c , there exists a δ ' > 0 such that if 0 < | y - c | < δ ' then

| f ( y ) - f ( c ) y - c - f ' ( c ) | < ϵ ' .

Now, by [link] , g - 1 is continuous at the point f ( c ) , and therefore there exists a δ > 0 such that if | z - f ( c ) | < δ then

| g - 1 ( z ) - g - 1 ( f ( c ) | < δ ' .

So, if | z - f ( c ) | < δ , then

| g - 1 ( z ) - c | = | g - 1 ( z ) - g - 1 ( f ( c ) ) | < δ ' .

But then,

| f ( g - 1 ( z ) ) - f ( c ) g - 1 ( z ) - c - f ' ( c ) | < ϵ ' ,

from which it follows that

| g - 1 ( z ) - g - 1 ( f ( c ) ) z - f ( c ) - 1 f ' ( c ) | < ϵ ,

as desired.

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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