# Tagged Questions

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### Roots Analysis. $f:[a,b]-->R$ continuous and for every x there is a y such as that$|f(y)|<|f(x)|/2$ .Show that exists ξ such that $f(ξ)=0$

$f:[a,b]-->R$ continuous and for every x there is a y such as that $|f(y)|<|f(x)|/2$ .Show that exists ξ such that $f(ξ)=0$
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### Continuity of function $f(x)=[x]\sinπx$

Examine the continuity of $f(x)=[x]\sinπx$ for $x \in\mathbb{R}$, where $[x]$ is the integral part of $x$.
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### Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
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### If a uniquness for all functions exist shouldn't there be uniquness to recursion?

What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example. Let's make a table of values for $^{x}2$ (0,1) (1,2) (2,4) ...
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### Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m$ is injective then $n \leq m$?

I had intended to restrict the image then $f:U \rightarrow f(U) \subset \mathbb{R}^m$ is bijective. Therefore $\dim f(U) = n \leq m$. That's right?
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### Election measurable in uniform continuity

Let $f:[0,1]\times [0,1] \rightarrow \mathbb{R}$ borel measurable such that for all $x \in [0,1]$ $f(x,-):[0,1] \rightarrow \mathbb{R}$ is continuous, in particular uniformly continuous. Then there ...
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### Show that $e^{-a|x|}$ does not belong to Schwartz space

Let $f : \mathbb R \to \mathbb R$ and $a > 0$ given by $f(x) = e^{-a|x|}$. Show that $f$ is rapidly decreasing and belongs to $L_1(\mathbb R)$, but not to $\mathcal S(\mathbb R)$. I had shown that ...
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### Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
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### $f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist

Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist. I think that if $f'$ ...
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### Example of a function?

$f$ is a discontinuous and bounded function defined on a closed set $C$. Also there exists a non-discrete closed subset in the image of $f$ such that it's inverse is open. Can you give an example ...
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### Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
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### Interesting question regarding elementary functions

I had this question at a test for a job interview and since and I didn't solve it. Some time later i still can't figure it out, so any insight is helpful. You need to write a function $f(x)$ such ...
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### Help me how to find the limit. In this case something is strange.

A fn(x) is a sequence of function. fn:[0,1]→R defined by fn(x) = n(1-x) × x^n. And I want to know limit of this function as n→∞. In my opinion we can see this like this n(1-x) × x^n. So as n→∞, ...
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### Prove the existence of exactly two maxima for a positive $L^1$ function

I have a function $f:\mathbb R \rightarrow \mathbb R^+$ with the following properties it is $L^1$ and $C^2$ it has one single extremum (maximum) at $x=0$ it is symmetric: $f(x)=f(-x)$. it is ...
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### Linearly approximating a curve [closed]

I'm trying to approximate y = x^3 with straight lines. One of the points at the end of two of the linear segments is (2,8). The linear segments are 22.5 units long. What are the coordinates of the ...
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### Extension of a holomorphic function in the disc

let $f$ be a continuos function in ${0<|z| \leq r}$ holomorphic in the inner and such that $f(z)$ is real for $|z|=r$. Prove that exist a function $g$ on $\mathbb{C} ^*$ such that $f(z) =g(z)$ ...
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### $f(z) = u(x,y) + i\cdot v(x,y)$ holomorphic in a connected open set $D$, such that $a\cdot u(x,y)+b\cdot v(x,y)=c$, is constant

Let $f(z) = u(x,y) + i\cdot v(x,y)$ be a holomorphic function in a connected open set $D$. If $a\cdot u(x,y)+b\cdot v(x,y)=c$ in $D$, where $a,b,c$ are real constants which are not all zero, why ...
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### A continuos and holomorphic function on $D^2$ that take pure imaginary values on $S^1$ is costant

Let $D := \{ |z| < 1\}$ and $f : \overline{D} \rightarrow \mathbb{C}$ be a continuos and holomorphic function on $D$ that take pure imaginary values on $\partial D$. Why $f$ is constant? From ...
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### An open map from $\mathbb{C} \rightarrow \mathbb{C}$ has open real and imaginary part?

If $f(z) :\mathbb{C} \rightarrow \mathbb{C}$ is an open map such that $f(z) = f_1(z) + if_2(z)$ where $f_1$ and $f_2$ represent respectively his real and imaginary part, we could say that both $f_1$ ...
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### Estimate error when using the result of an expanded equation

I would like to know how to deal with the error term in expanded expression. For example consider the function $\displaystyle f(x)=A\text e^{-(x+\lambda)^2}+B\text e^{-(x-2\lambda)^2}\;,$ where ...
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### How many extrema has $f(x)$ from $L^2$ if it is not a polynomial?

Is it possible to say how many extrema the $L^2$ function $f(x)=A\text e ^{-(x+\lambda)^2} - B\text e ^{-x^2} + C\text e ^{-(x-\lambda)^2}$ has on whole $\mathbb R$? Here $A$, $B$, $C$ and ...
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### How prove $f(x)$ is monotonous , if $f'(x)=g[f(x)]$

Question: Let $f(x)$ be a derivative, and there exsit $g(x)$ be such that: $$f'(x)=g[f(x)]$$ Show that $f(x)$ is monotonic. This problem is from Xie Hui Min analysis problems book in china ...
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if i have $\displaystyle \lim_{|u|\rightarrow 0}\frac{f(t,u)-a|u|^{\tau-2}u}{u}=0$ how to prove that $\displaystyle \lim_{|u|\rightarrow 0}\frac{f(t,u)}{|u|^{\tau-2}u}=a$ such that $\tau\in (1,2)$ I ...
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### Where to stop a taylor expansion of a function of more than one variable?

I tried to taylor expand a function of three variables $f(x,y,z)$ around $y=0,z=0$ but I don't find a way to decide where to stop the expansion. Usually when one expands a function of one variable ...
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### To calculate the limit :

$$\lim_{n\rightarrow\infty}{n^2}(\arctan\frac{a}{n}-\arctan\frac{a}{n+1})$$ I used the formula $\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$,but it just doesn't work. Waiting for your help...
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### Contraction map

I have a general question about the properties of contractive/non-contractive maps. Assuming I have a map $F$ which I know is not a contraction. Can it map a ball back to itself, i.e. for some ...
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### right derivative of a continuous function

Let $f:(a,b)\longrightarrow \mathbb{R}$ be continuous. Suppose $D_+f(x)=\lim_{h\to 0+}\frac{f(x+h)-f(x)}{h}\geq 0$ for any $x\in (a,b)$. Prove that $f(x_1)\geq f(x_0)$ whenever $x_1\geq x_0$. How to ...
Does there exist real functions $f, g\in C^1[-1,1]$ such that $$\det\left(\begin{array}{cc}f &g \\ f'&g'\end{array}\right)\equiv0 \qquad \det\left(\begin{array}{cc}\int_{-1}^1f^2\,\mathrm ... 1answer 18 views ### Show that the following functi0n is bounded Let f: \mathbb{R} \rightarrow \mathbb{R} be a continuous function with f(0)>0 and$$\lim_{x \to \infty} f(x)= \lim_{x \to -\infty} f(x)=0 $(i)$ Show that $f$ is bounded. $(ii)$ Let ...
A theorem in calculus: Let $f:\mathbb{R}^2\longrightarrow \mathbb{R}$. If $\lim_{(x,y)\to (a,b)}f(x,y)=A$ exists (either finite or $\infty$) and there exists $\epsilon>0$ such that for any ...