# 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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### Proving that $A \mapsto \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$ is an inner measure

Let $(X,\Sigma, \mu)$ be a measure space and define $m: 2^X \to [0,\infty]$ by $m A = \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$. Show that $m$ is an inner measure. There are $4$ ...
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### Is it possible / allowed to use L'Hôpitals rule for products?

In our readings, we had L'Hôpitals rule and defined it like that: $\lim_{x\rightarrow x_{0}}\frac{f'(x)}{g'(x)}$ Because we had it in our readings, we are allowed to use this to find limit of ...
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### Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$

The question is as follows: Given: (1) function $f: U \subset \mathbb R^n ==> \mathbb R$ (2) $U$ is open and convex set (3) $f \in C^1$ in $U$ Goal: Show that $f$ is ...
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### $\text{ Proving }\; A \subseteq \Bbb R \text{ A is bounded above} \Rightarrow A^c \text{ is not?}$

Prove: Let $A \subseteq \Bbb R$. Prove that if $A$ is bounded above, then $A^c$, the complement of $A$ is not bounded above. $A^c =$ those element of the universe that are not in A. $\Bbb R =$ ...
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### Differential $\mathrm{d}^2f$ of implicit function $F(x,y,z)=xyz-x-y-z=0$

Determine the differential $\mathrm{d}^2f$ of the implicit function defined as $z=f(x,y)$: $$F(x,y,z)=xyz-x-y-z=0$$ So in fact of the implicit function I have to use the implicit function ...
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### Determine the range of f(x)=(sinx)/x

I am having trouble understanding the solution to this question. ''Determine the range of the following function: $f(x)$ = $(1$ $if$ $x=0)$ or (${\sin x\over x}$ if $x$$\neq$$0$) where the domain ...
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### Upper hemicontinuity of a correspondence

I would like to know whether the following correspondence is upper hemicontinuous: $$C(x)=\begin{cases} 1, & (f(x)>0) \\ [0,1], & (f(x)=0) \\ 0, & (f(x) < 0) \end{cases},$$ ...
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Let $\psi \in C_0^{\infty}(\mathbb{R}^3)$. How to prove (or where I can find this proof) that $$\int_{\mathbb{R}^3}\frac{1}{4r^2}|\psi(x)|^2d^3x\le \int_{\mathbb{R}^3}|\nabla\psi(x)|^2d^3x$$ ? ...
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### Using Rolle's Theorem to prove that f'(c)=rf(c). [on hold]

Suppose that the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)=0$. Prove that for each real number $r$, there is some $c$ on $(a,b)$ such that $f'(c)=r\cdot f(c)$. ...
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### Definite integral of $\sqrt{\frac{1}{\cos^2(x)}}$

I've got problems with this integral: $$\int_0^{\frac{\pi}{4}} \sqrt{\frac{1}{\cos^2(x)}} \, \mathrm{d}x$$ First I substitute $x=2\arctan(x)$ but this leads nowhere. Any hints for solving?
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### How to solve this Sturm Liouville problem?

$\dfrac{d^2\phi}{dx^2} + (\lambda - x^4)\phi = 0$ Would really appreciate a solution or a significant hint because I could find anything that's helpful in my textbook. Thanks!
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### Requirements for the Functional Equation $\sum_{i=1}^{n}f_i(x)g_i(y)=0$

Consider the following functional equation $$\sum_{i=1}^{n}f_i(x)g_i(y)=f_1(x)g_1(y)+f_2(x)g_2(y)+\cdot\cdot\cdot+f_n(x)g_n(y)=0 \tag{1}$$ where $f_i(x)$ and $g_i(y)$ are arbitrary functions. ...
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### particle velocity variable with time [on hold]

Consider two fixed points $A$ and $B$. At $A$, you have a receiver and at $B$ you have a transmitter. $B$ continually emits particles towards $B$ at a constant rate. But the particle velocity is ...
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### Integrate function by partial derivative

I'm searching a $\phi(x,t)$ solution of a pde cauchy system, with $x\in[-1,1],t\in[0,T]$ I am able to know: a) $\phi(x,0)=-cos\left(\pi\left(x-0.85\right)\right)$ b) $\phi_x(x,t)$, $\forall t,x$ (...
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### Analyze the monotony of this function - I almost got it

Analyze the monotony of $f(x) = (1+x+\frac{1}{2}x^{2})e^{-x}$ To analyze the monotony of a function, you can build the first derivation of the function, equalize it with $0$ and if the (derivative) ...
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### Equality case in the Prékopa-Leindler inequality.

in the paper 'Remarks on the conjectured log-Brunn-Minkowski inequality' by C. Saraoglou, the author uses the result (Lemma A. 3.) about the equality case in the Prékopa-Leindler inequality. For the ...
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### Value of tan(pi/2) [duplicate]

I understand that this is a very stupid question but I'm not getting the answer. At $x=\pi/2$, what is the value of $tan(x)$? Should it be $-\infty$ or $+\infty$? Text tells it to be $+\infty$. But ...
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### Multivariable implicit function - Jacobi Matrix

Find the derivate $f',f''$ of the implicit function $z=f(x,y)$ defined by the following equation: $$F(x,y,z)=x^2+y^2+z^2-a^2=0$$ So the first step to build the Jacobi-Matrix $f'$ lead me to ...
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### Construction of a measure space from some weird functional

Here is the complete problem I am trying to solve, but currently, I am just interested in proving that $\Sigma$ is a $\sigma$-algebra. Let $X$ be a set and $\phi: 2^X \to [0, \infty]$ be a ...
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### Is value of $\pi = 4$?

What is wrong with this? SOURCE
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### Are derivatives actually bounded?

Suppose you a function $f$ which is differentiable, with the property that $$f^{(n)} (0) = (n!)^2$$ And in general $$f^{(n)} (a) = O((n!)^2)$$ For any $a \in \mathbb{R}$. This function ...
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### A question about Inverse Function theorem

The Inverse Function theorem: Let A be open in $R^n$; let $f: \to R^n$ be of class $C^r$. If $Df(x)$ is non-singular at the point $a$ of $A$, there is a neighborhood $U$ of the point $a$ such that $f$ ...
### Proving $f(x)=x^2 +3x+1$ is continuous on $(0,1)$ by definition
Def: A function is continuous if f : I → R at c ∈ I means that for every ε > 0 there exists a δ > 0 such that for all x ∈ I, |x-c| < δ => |f(x) - f(c)| < ε Prove that $f(x)=x^2 +3x+1$ is ...