# 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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### Upper integral of a production funcion, can i assume this about the supremum?

Let $f: A \rightarrow \mathbb{R}$ $g: B \rightarrow \mathbb{R}$ be bounded and non negative functions at blocks $A$ and $B$. Define $\phi: A\times B \rightarrow \mathbb{R}$ as $\phi(x,y) = f(x)g(y)$. ...
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### $f: \Bbb R^2 \to \Bbb R$ whose partials exist. Show: $\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$

Let $f: \Bbb R^2 \to \Bbb R$ be a function whose partial derivatives exist. Now i have to show: $$\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$$ Any tipps on how ...
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### Show that integral is analytic

Let $h:[0,\infty)$ be an integrable function. Prove that the function $$g(z)=\int_0^\infty h(t)e^{tz}\,dt$$ is analytic on $\{z=x+yi:x<0,y\in\mathbb{R}\}$. How do I start for this question? I ...
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### Dilation convergence in L^1

Below is a question, which I asked before, from Stein's Real Analysis. I've provided a partial solution, which I think it's pretty along the lines of what needs to be done, however, I have no ...
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### $f: \mathbb R^2 \to \mathbb R$ is differentiable when both partials exist and one is continuous

I'm trying to solve the following exercise: Let $f:\mathbb R^2 \to \mathbb R$ a continuous function whose partials exist everywhere in $\mathbb R^2$. Show that $f$ is ($\mathbb R$-)differentiable ...
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### would changing the lower limit of a power series affect radius of convergence

When we change the lower limit of a power series by any finite quantity, would it increase or decrease radius of convergence or no change? Clarification of terminology: There might be confusion about ...
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### Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover.

I really need help with this question: Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover.
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### Let $\phi:(0,1)\to \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ given by $\phi(t)=A(tx)$, then $\phi'(t). h=(A'(tx). x). h$ or $(A'(tx). h). x$?

Let $U$ be an open ball centered in $0$ in $\mathbb{R}^m$. Given $\phi:(0,1)\to \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ be defined by $\phi(t)=A(tx),$ where $A:U\to \mathbb{R}^n$ and $x\in U$, which ...
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### Does being Nonempty Compact Set on $\mathbb{R^+_2}$ imply being Convex set?

Look at the domain of a function $y=x-2$ where $x\in\mathbb{R_+}$. Then, the triangle produced by x and y-intercepts is bounded and closed. So it is compact. Suppose it is also nonempty. Does this ...
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### Is a surjective mapping of R2 to itself with full rank derivative everywhere necessarily injective?

If $f:\mathbb R^2\rightarrow\mathbb R^2$ has rank 2 derivative everywhere, then by the inverse function theorem it is locally injective. If it is surjective, is it then necessarily globally injective ...
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### Question about $N-$functions

If $\Phi$ is a $N-$function, and $(u_n)$a sequence from $W^{1,\Phi}(\mathbb{R}^N)$ such that $u_n\rightarrow u$ in $W^{1,\Phi}(\mathbb{R}^N)$ Can we prove that $$\Phi(|u_n|)\leq \Phi(|u_n-u|+|u|)$$ ...
I deleted my last question because there was a huge mistake inside. Given: $R$ is the radius of convergence of $\sum_{n=0}^{\infty} a_{n}x^{n}$, also suppose that $\lim_{n\rightarrow \infty} \left | \... 1answer 29 views ###$p(x) \in \mathbb R[x]$be a polynomial of odd degree ,$n>1$be an integer , then is the function$A \to p(A)$surjective on$M(n,\mathbb R)$? Let$p(x) \in \mathbb R[x]$be a polynomial of odd degree ,$n>1$be an integer , then is the function$f: M(n,\mathbb R) \to M(n, \mathbb R)$defined as$f(A)=p(A) , \forall A \in M(n,\mathbb R)$... 1answer 39 views ###$p(x) \in \mathbb R[x]$be non-constant polynomial ,$n>1$, the function$A \to p(A)$is surjective on$M(n, \mathbb C)$? Let$p(x) \in \mathbb R[x]$be a non-constant polynomial and$n>1$, then is it true that the function$f:M(n,\mathbb C) \to M(n, \mathbb C)$defined as$f(A)=p(A) , \forall A \in M(n, \mathbb C)...
Let, $\mathbb{P}$ be the set of prime numbers. We know that $\mathbb{P}$ has a bijection with set of Naturals $\mathbb{N}$. That is, $\mathbb{P} \stackrel{}{\longleftrightarrow} \mathbb{N}$. Again, ...