0
votes
1answer
14 views

Is the principal value of Argument differentiable at every nonnegative nonzero number?

How do i show that argument is continuous at points except its branch cut? I posted a question to ask whether the principal value of Argument $Arg:\mathbb{C}\setminus \{0\}\rightarrow (-\pi,\pi]$ is ...
0
votes
1answer
35 views

Continuity of a complex function $f$ at $a$ implies $\lim \limits_{z\to a}(z-a)f(z)=0$

Assume the complex valued function $f$ of a complex variable is continuous at $a\in\mathbb C$. How can we see that $$\lim_{z\to a}(z-a)f(z)=0$$
1
vote
1answer
55 views

How to show that a complex-valued function is uniformly continuous?

should a function be uniformly continuous in both arguments if it should be uniformly continuous as a complex-valued function. For example how can I proove that ...
1
vote
2answers
226 views

Find a function that satisfies the condition.

Let $\epsilon > 0$ be fixed and $t$ a variable that takes values in the universal covering space of ${\mathbb{C} \setminus \{0\}}$. Find a continuous function $f(s$) such that $$|t \log t| = |t| ...
1
vote
1answer
33 views

Show continuity and holomorphism for a function

Let $A = \left\{ z \in D_r \big| \; Im(z) \geq 0 \right\}, f : A \rightarrow \mathbb{C}$ continuos on $A$, holomorphic on the inner of $A$ and real-valued for $\left]-r,r\right[$. Let further be ...
3
votes
1answer
38 views

Relation between continuity of $f$ and analyticity of $f(z)^8$

If $f(z)$ is continuous on some domain $D$ and $f(z)^8$ (the function to the eighth power, not the eighth derivative) is analytic, then why does this imply that f is analytic on a neighborhood of each ...
0
votes
1answer
25 views

Continuity of Complex function and restrictions

I am trying the following question but am stuck at finding the restriction: Prove that $f(z)=1/z^2$ is continuous at $z_0= 1+2i$ Solution: I am trying the use epsilon-delta proof and got it down to: ...
2
votes
1answer
47 views

Let C be a circle. Show that the only subset of C homeomorphic to a circle is C itself

I am trying to answer the question stated in the title. The hint in my book says to realize that for any z on the circle C{z} is still connected. I believe I can deal with case that shows that C{z} ...
1
vote
2answers
65 views

Proof that a complex function is continuous at $z=0$

Given the function $f\colon \mathbb{C}\to\mathbb{C}$ by $f(z)=\begin{cases} \frac{xy(x+iy)}{x^2+y^2} & \text{if } z\neq 0\\ 0 & \text{if } z=0 \end{cases}$ with $z=x+iy$. How do I ...
0
votes
1answer
31 views

The Group of Complex Continuous Functions?

Let $C(\mathbb{C},\mathbb{C})=\{f:\mathbb{C} \rightarrow \mathbb{C}\,|\,f $continuous $\}$ be the set of all continuous functions from the complex plane to itself and consider the composition ...
2
votes
4answers
178 views

Is $f(z)=\bar{z}$ continuous?

I have $z\in \mathbb{C}$, is $f(z)=\bar{z}$ continuous on the whole complex plane? Note that $\bar{z}$ is the conjugate of $z\in \mathbb{C}$ I was thinking that if $z$ is on the real line, then ...
0
votes
2answers
46 views

Continuity of $f(z)=u(x,y)+iv(x,y)$

If $u(x, y)$ and $v(x, y)$ are continuous (respectively differentiable) does it follow that $f(z) =u(x, y) + iv(x, y)$ is continuous (resp. differentiable)? If not, provide a counterexample. This ...
1
vote
2answers
350 views

Proving a complex function is continuous.

I've recently started complex analysis but I have very little background in complex numbers and to make sure I don't fall behind I'm doing some extra exercises one of which is Show $f$ is continuous ...
0
votes
1answer
44 views

limit of continuous function in complex plane

When I have a function, say $v$, continuous at $z_0$, then $\lim_{z\to z_0}v(z)=v(z_0)$. Does that imply that $\lim_{z\to z_0}iv(z)=iv(z_0)$, where $i$ is the imaginary number. Also, if I have u as ...
1
vote
3answers
158 views

let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume $f(z) = f(2z)$, prove that f is constant

$f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume that $f(z) = f(2z)$ for all $z \in \mathbb{C}$. Prove that f is constant... Then we are supposed to use this result to ...
1
vote
2answers
38 views

Extending an holomorphic function

Let $D \subset \mathbb{C}$ be a disc. Is there a function $$ f \in H(\mathop D\limits^{\circ} ) \cap C(\overline{D}) $$ such that , for every open set $A \supset \overline{D} \ $, $f \notin H(A) \ \ ...
7
votes
4answers
246 views

On continuity of roots of a polynomial depending on a real parameter

Problem Suppose $f^{(t)}(z)=a_0^{(t)}+\dotsb+a_{n-1}^{(t)}z^{n-1}+z^n\in\mathbb C[z]$ for all $t\in\mathbb R$, where $a_0^{(t)},\dotsc,a_{n-1}^{(t)}\colon\mathbb R\to\mathbb C$ are continuous on ...
1
vote
1answer
40 views

Mean Value Property for Continuous Complex Functions

Suppose I have an open set $U$ in the complex plane and a function $g$ that is continuous on $U$. Let $C(z_0$$,r)$ be a circle fully contained in $U$ of radius $r$ whose center is $z_0$. I know ...
2
votes
1answer
128 views

To what extent is a function that is analytic on the unit disk determined by its boundary values?

Suppose we have a function that is analytic on the open unit disk. Suppose we have a continuous function on the boundary of the disk that maps each point on the boundary of the disk to its conjugate. ...
5
votes
1answer
70 views

Continuity of Green's function

Suppose $\Omega \subset \mathbb C$ is a region (open and connected set) and let $$g(z,z_0)=G(z,z_0)-\log|z-z_0| $$ be its Green's function with pole at $z_0 \in \Omega$. Here $G(z,z_0)$ is the ...
1
vote
1answer
63 views

Show that the the multipliction and inverse operations on the quaternion unit sphere are continuous

This is a bit of a tricky question, we define the real Quaternions as: $$H=\left\{ a+bi+cj+dk\mid a,b,c,d\in\mathbb{R}\right\}$$ With the rule that: $$ij=-ji=k\:,\: jk=-kj=i\:,\: ki=-ik=\, j\;,\: ...
2
votes
1answer
89 views

Showing continuity

Let $|f(x)|\le \dfrac{A}{1+x^2}$ for all $x$ and some $A$ (to ensure that $\int_{-\infty}^\infty f(x)\rm{d}x$ makes sense). I would like to show that $$g(z) = \int_{-\infty}^t f(x) e^{-2 \pi i z ...
1
vote
2answers
80 views

Does a connected space implies continuity and vice versa?

I was thinking about this question when we gone over limit and continuity in my complex analysis class. Let a region $R$ be the image of a function $F$, if I can prove that $R$ is a connected space ...
2
votes
1answer
147 views

Show that $\sigma^{-\lambda(x) - 1}$ is continuous on $(0,1)$.

Let $V$ be an open connected subset of $\mathbb{C}$ and $A(V)$ be the set of all (complex-valued) analytic functions on $V$. If $\lambda \in A(V)$ with $\Re \lambda(x) < 0$ for all $x \in V$ , ...
0
votes
0answers
111 views

Cauchy theorem for “closed triangular paths” and complex continuity

I am studying for my calculus exam and I have some problems related to the proof of the Cauchy theorem for “closed triangular paths”. This version actually goes by its own name the “Goursat” theorem. ...
1
vote
1answer
154 views

Check my work: $\lim a_n = 0 \Rightarrow \lim \sqrt{a_n} = 0 $? (for $a_n$ positive)

I'm trying to prove, as "properly" as possible the following:$$\left[ \lim z_n = z \right] \iff \left[ \lim x_n = x \quad \wedge \quad \lim y_n = y \right]$$ where $z_n = x_n + i y_n$ and $z=x+iy$. ...
1
vote
1answer
112 views

Morera's theorem of entire function

For each fixed $n$, show that $$f_n(z)=\int_1^nt^{z-1}e^{-t}dt$$ is an entire function of $z$. From Morera 's theorem: If a continuous, complex-valued function $f$ in a domain $D$ that ...
4
votes
1answer
60 views

An analytic function is onto

All sets are subsets of $\mathbb{C}$. Suppose $f: U \to D$ is analytic where $U$ is bounded and open, and $D$ is the open unit disk. Now suppose we can continuously extend $f$ to $\bar{f}: \bar{U} ...
8
votes
3answers
221 views

Continuous extension of analytic functions

Is it possible to prove the following statement or is there a counter-example: Let $H=\{y>0\}$ be the upper half plane in the complex plane. If $f$ is an analytic function on $H$ and its real part ...
0
votes
2answers
155 views

How to show that $z^4$ is not uniformly continuous?

$f$ maps $z$ in $\mathbb C$ to $z^4$ in $\mathbb C$. How do I show that this function is not uniformly continuous?
0
votes
1answer
48 views

Continuous complex funtion

I have this function $$F(z)=\frac{1}{\alpha-i\sqrt{z}}$$ with $\alpha>0$ and the determination of the square root with $\Im z>0$. I have to study its continuity in the set $$A=\lbrace z|a\leq\Re ...
1
vote
1answer
102 views

How can I study the continuty of this function?

Let $f\in L^2(\mathbb{R}^3)$ with compact support; is the function $$F(z)=\int_{\mathbb{R}^3}dx\bigg(f(x)\frac{e^{i\sqrt{z}|x|}}{4\pi|x|}\bigg)$$ continuous in the set $$Q=\lbrace{z: \Re z\in [a,b], ...
0
votes
0answers
188 views

Analytic functions over an open subset vs continuous functions over its closure

Let $H(G)$ denote the set of analytic functions over an open subset of the complex plane $G$. Let $A(G)$ denote the collection of continuous functions over the closure of $G$. Can you provide an ...
0
votes
1answer
75 views

Extended functions continuous at $z=(0,0)$

There are 4 functions: $$\frac{Re(z)}{|z|},\frac{z}{|z|},\frac{Re(z^2)}{|z|^2},\frac{zRe(z)}{|z|}$$ I need to determine which of these functions can be defined at $z=0$ in such a way that the ...
1
vote
3answers
150 views

Why is it clear from this formulation that f is continuous wherever it is holomorphic?

Hi I am new on here so not sure if this is right place to post but quick and presumably easy question: So holomorphic at a point $z_0 \in \Omega$ is defined as the limit as $h\rightarrow 0$ of ...
0
votes
1answer
660 views

Continuity of Complex Function

Since we can represent complex functions as $f(z) = u(x,y) + iv(x,y)$, under what conditions do we know that $f(z)$ is continuous? For example, if $u(x,y)$ and $v(x,y)$ are both continuous can we ...
0
votes
0answers
95 views

Isolation of zeros in the case of univariate analytic functions expressed as a bivariate function.

We know that the zeros of an analytic non-constant function are always isolated. A proof is here. Let $L(v)$ be an analytic function in $v$, where $v\in\mathbb{R}$. Let us write $L(v) \equiv L(v,p)$ ...
1
vote
0answers
57 views

Lower-bounding the distance between zeros of a continuous function

Consider a continuous function of the form: $L(v) = \sum_{i = 0}^{m}[vA_{i} - B_{i}]p^{i}$ where $p$ is the root of the polynomial equation: $vf(p) - g(p) = 0$ with $f(p)$ and $g(p)$ being two ...
2
votes
1answer
290 views

A Continuity Argument in the Proof of Rouche's Theorem

In Greene and Krantz's Function Theory of One Complex Variable, the proof of Rouche's theorem involves the following continuity argument. Let $f,g\colon U \to \mathbb{C}$ be holomorphic from an ...
0
votes
1answer
137 views

Proving Continuity in Several Complex Variables

So I don't have a whole lot of experience in general proving continuity for multivariable functions, and I want to make sure I'm going about things correctly. Prove that the function ...
1
vote
1answer
56 views

Question about Continuity of Path Integrals

I have this continuous function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined on an open set $\Omega$. I also have a family of identical smooth curves up to translation ...
3
votes
1answer
285 views

continuous function from $[0,1]$ to $\mathbb{C}$

Let $T=\{z\in \mathbb{C} :|z|=1\}$ and $f:[0,1]\rightarrow \mathbb{C}$ be continuous with $f(0)=0$ and $f(1)=2$. I need to show that there exists at least one $t_0\in [0,1]$ such that $f(t_0)\in T$. ...