# Tagged Questions

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### 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 ...
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### 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 $$\lim_{z\to a}(z-a)f(z)=0$$
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $L(v) \equiv L(v,p)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
### 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 $f(t_0)\in T$. ...