# If f(z) is 1-1, then f'(z) is not zero.

• Bacle
In summary: This is why I think that if f is 1-1, the assumption that f'(z) is not zero is needed ( but I am not expert in topology, so I don't know if this assumption is intrinsically in contradiction with 1-1, but I think not).In summary, we are trying to rigorize the result that if f is one-to-one in a region R, then f'(z) is not zero in R. One approach is to assume the contrary and show that it leads to a contradiction. Another approach is to use the inverse function theorem to show that if f'(z) is zero, then the inverse of f is not differentiable, which contradicts the
Bacle
Esteemed Analysts:

I am trying to rigorize the result that if f is 1-1 in a region R, then f'(z) is not zero in R.

This is what I have: Assume, by contradiction, that f'(zo)=0 for zo in R. Then

f can be expressed locally as :

f(z)=z^k.g(z)

for g(z) analytic and non-zero for some open ball B(zo,r)-{zo}

From this, we have to somehow use the fact that z^k is k-to-1, contradicting the

assumption that f is 1-1.

I don't know if we can use the fact that an open ball is simply-connected to

define a branch of log, from which we can define a branch of z^k, and then

conclude with the contradiction that f(z) is not 1-1.

Any suggestions for rigorizing?

Thanks.

One argument goes as follows:

WLOG z0=0 and f(0)=f'(0)=0. So f(z)=zkg(z) for some integer k>1, g analytic and nonzero in a n'hood of 0. So we can take a kth root of g, say g(z)=h(z)k. So f(z)=(zh(z))k. Now zh(z) is a nonconstant holomorphic function (if it's constant, it's zero, so f is 0 identically) and so it's an open map. So the image of any open n'hood of zero is open, and hence contains some open disc around 0. This must include points different by a factor $\exp(2\pi i/k)$, from which f is not injective.

WWGD
Thanks, henry_m:

Yes, that is along the lines I was thinking, but I did know how to justify the
existence of a holomorphic k-th root in a ball about 0; I was thinking of using
the fact that a ball is symply-connected, so that we can define a branch of log,
from which we can define a root, but there may be other ways of showing the
existence of a k-th root in B(zo,r)?

Are you worried about the step of finding h given g? You don't quite need to use the fact that a disc is simply connected, basically because we only need to be able to work in a tiny disc around z0.

In more detail:

Pick some disc around g(z0) not containing zero. It should be clear that we can define a branch on the logarithm in this disc. Then h(z)=exp(log(g(z))/k) defines h on the preimage of this disc, which contains z0.

Question: Consider f(z)=z^3. Then, isn't f(z) 1-1 and f'(0)=0?

3.1415926535 said:
Question: Consider f(z)=z^3. Then, isn't f(z) 1-1 and f'(0)=0?

Not in the complex plane. For instance, let $\zeta = e^{2 \pi i/3}$. Then $\zeta$ is a primitive third root of unity, so
$$1 = \zeta^3 = (\zeta^2)^3 = 1^3.$$

So f is actually 3-to-1. You are correct for the real line, though, f(x) = x3 is one-to-one on the reals.

As for the original question:

If f is one-to-one on R, then it has an inverse $f^{-1} \colon f(R) \to R$. I think the derivative of the inverse is given by
$$(f^{-1})'(w) = \frac{1}{f'(z)}$$
where f(z) = w. So f'(z) can't be 0 or (f-1)' will blow up at f(0).

Does this work? I might have forgotten a condition...

spamiam said:
As for the original question:

If f is one-to-one on R, then it has an inverse $f^{-1} \colon f(R) \to R$. I think the derivative of the inverse is given by
$$(f^{-1})'(w) = \frac{1}{f'(z)}$$
where f(z) = w. So f'(z) can't be 0 or (f-1)' will blow up at f(0).

Does this work? I might have forgotten a condition...

The problem with this argument is that you assume that the inverse is analytic, which must be proved.

The normal complex inverse function theorem assumes nonzero derivative, and proves existence of a continuous inverse, and then analyticity of the inverse and the formula for the inverse. You are trying to use the formula without the first assumption. To see what goes wrong if we try to use the formula here, the argument for the last part goes like this:

Let g be an inverse for f, and w, w' be in the domain of g and not equal. Let z=g(w), z'=g(w'). Then z and z' are not equal, and:
$$\frac{g(w)-g(w')}{w-w'}=\frac{z-z'}{f(z)-f(z')}$$
Now as w tends to w', z tends to z' (continuity of g needed) and so the RHS tends to 1/f'(z'), which proves what we wanted; g is differentiable with g'(w')=1/f'(z').

BUT if we haven't assumed that f' is nonzero, we have proved only that the limit, and hence the derivative of g, does not exist if f' vanishes somewhere. There is no contradiction since we haven't proved that the inverse is differentiable.

Spamiam:

The problem is that I don't know how to tell if 1/f'(z) is analytic or not.

My approach is this, assuming that if f is 1-1, then f'(z) is not 0:

The Jacobian matrix of an analytic function is an antisymmetric Jacobian,

with the entries the partials of U,V with respect to x,y respectively. Since

f'(z) is not zero, the determinant J(f):=U_x^2+U_y^2 is itself not zero, so J(f)

is invertible. You can then show that the form of the inverse J(f)^-1 is the same

as that of J(f), i.e., J(f)^-1 is also antisymmetric, and contains the partials

of f^-1(z) with respect to x,y, and these satisfy Cauchy Riemann, by the

symmetry of the inverse matrix.

Ah okay, I thought I was forgetting something. Thanks for the correction.

Actually, Spamiam, I think your statement is true if f(z) is real ( tho the names are changed

the innocent :) ):

If f(z)=U+iV is analytic, then:

Ux=Vy

Uy=-Vx

But , given 1/f'(z)= f'(z)^/f'(z).f'(z)^ (with ^ complex conjugation )

=U-iV/|f'(z)| , forces Ux=Vy=-Vy , so that Vy is constant; same for other component functions.

## 1. Why does f(z) being 1-1 mean that f'(z) is not zero?

When a function is 1-1 (injective), it means that each input value (or domain) has a unique output value (or range). This also means that no two input values can have the same output value. If f'(z) was zero, it would mean that the function has a constant slope and therefore, multiple input values could have the same output value, contradicting the definition of a 1-1 function.

## 2. Can a function be 1-1 and have f'(z) equal to zero at certain points?

Yes, it is possible for a function to be 1-1 and have f'(z) equal to zero at certain points. This can happen when the function has a horizontal tangent at those points, but still maintains a unique output value for each input value.

## 3. How does the graph of a 1-1 function differ from a non-1-1 function?

The graph of a 1-1 function will pass the horizontal line test, meaning that no horizontal line can intersect the graph at more than one point. This results in a graph that does not have any vertical lines of symmetry. In contrast, a non-1-1 function may have intersecting points with a horizontal line, resulting in a graph with vertical lines of symmetry.

## 4. Are there any exceptions to the rule that f(z) being 1-1 means f'(z) is not zero?

No, there are no exceptions to this rule. The definition of a 1-1 function inherently requires that f'(z) is not equal to zero in order to maintain a unique output value for each input value.

## 5. How does the derivative of a 1-1 function relate to its inverse function?

The derivative of a 1-1 function is related to its inverse function by the inverse function theorem. This theorem states that if a function is differentiable and its derivative is never zero, then its inverse function is also differentiable. This means that the inverse function will also have a non-zero derivative, ensuring that it is also a 1-1 function.

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