# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### Relating real integrability and integration over the unit disk.

Suppose we have a function $f:\mathbb{R} \rightarrow [0,\infty)$, such that $\int_0^1f < \infty$. Is it always true that $\int_\mathbb{D} f(|z|)dA < \infty$ ? $dA$ is area measure. If so, ...
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### strictly increasing functions and bijection

$f$ is strictly increasing \begin{align} & \rightarrow f^{-1} \text{is strictly increasing}\tag {i} \\ & \rightarrow f^{-1} \text{is strictly decreasing}\tag {ii} \\ & \rightarrow ...
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### Limit of a continuous real-valued function on $(0,\infty)$ as $x\to\infty$.

Let $f:(0,\infty)\to\mathbb R$ be a continuous function. Suppose that $f$ satisfies $$\lim_{x\to \infty}\left( f(x+1)-f(x)\right) = 0.$$ Show that $\lim_{x\to \infty }\dfrac{f(x)}{x} = 0.$ I tried ...
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### Example to prove that $C_c(\mathbb{N})$ is not a Banach space for the uniform norm?

I know The space $(C_c(\mathbb{R}), \lVert\,\cdot\,\rVert_u)$ is not complete, so it is not a Banach space. But for $X = \mathbb{N}$, why is $C_c(X)$ not a Banach space? Can a sequence of functions ...
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### Find a sequence of differentiable functions $\{f_n\}$ on $[0,1]$ such that $f_n \to 0$ uniformly, but $\{f_n'(1/2)\}$ does not converge to $0$. [duplicate]

Find a sequence of differentiable functions $\{f_n\}$ on $[0,1]$ such that $f_n \to 0$ uniformly, but $\{f_n'(1/2)\}$ does not converge to $0$. I have tried to find an example which follows above ...
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### How to prove function $f(x,y)=\frac{1}{xy}$ is not uniformly continuous?

Here I consider uniform continuity of functions in $\mathbb{R}^n$. Take a function of two variables for example. We said that $f(x,y)$ is uniformly continuous if for any $\epsilon>0$, we can find ...
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### Completion of the rational numbers

I am preparing to embark on the completion of $\mathbb{Q}$ with respect to a p-adic absolute. However, I thought I should start with the completion of $\mathbb{Q}$ with respect to the usual absolute ...
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### A limit problem with a background

Could you show me $$\mathop {\lim }\limits_{x \to 0} \left( {1 + \sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n}{{\left( {\frac{{\sin nx}}{{nx}}} \right)}^2}} } \right) = \frac{1}{2}.\tag{1}$$...
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### function and derivative converging to constants implies derivative is 0

I've been working on this problem from my textbook. Suppose that $f$ is differentiable on $(a,\infty$). Show that if $f'(x) \to L$ as $x \to \infty$, then $\frac{f(x)}{x} \to L$ as $x \to \infty$. ...
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### Prove that if $2^{k_1}l_1=2^{k_2}l_2$, then $k_1=k_2$ and $l_1=l_2$.

Suppose $k_1,k_2,l_1,l_2$ are natural numbers such that $l_1$ and $l_1$ are odd. Prove that if $2^{k_1}l_1=2^{k_2}l_2$, then $k_1=k_2$ and $l_1=l_2$. First, let $l_1 = 2m_1 +1$ and $l_2=2m_2+1$ ...
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### Finding smallest closed affine subspace

Given the space $H=\mathbb{L}^2(0,1)$, and the subspace $K$ of $H$ defined by: $$K=\left\{f\in H: f\geq 0\right\},$$ I would like to determine the smallest closed affine subspace of $H$ which contains ...
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### Take the derivative of the norm? can I do this?

Here is a quick question to the math community. Is it possible to take the derivative of the euclidean norm? For example, if $f(x)=\|x\|$ then $f'(x)= \|1\|=1$ and $f''(x)=0.$ I know it is ...
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### Closed, open and non-empty subset of $\Bbb R^n$ equals $\Bbb R^n$ [closed]

$A$ is a subset of $\mathbb{R}^n$ closed , open and non-empty . Prove that $A = \mathbb{R}^n$.