Elementary questions about functions, notation, properties, and operations such as function composition.

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Continuous functions on compact group and uniformity

If $G$ is a compact abelian group and $f\in C(G)$. Then $\forall \epsilon >0$,there exists an open neighbourhood $U$ of $0\in G$, such that $\forall g\in G , \forall u_1,u_2\in U$, we have ...
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0answers
19 views

$f(x) = 2x \mod 1$ not equal to zero for all $x$?

If any number $\mod 1$ is zero, then how can $f(x) = 2x \mod 1$ be a Baker's map? For any $x\in \mathbb{R}$, shouldn't $f(x)=0$?
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1answer
27 views

Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
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2answers
23 views

Constancy of an integral function

Fix some $\ell\in\mathbb{R}^+$. Say that $f:\mathbb{R}^2\to\mathbb{R}_{\geq0}$ and $\mu:\mathbb{R}\to\mathbb{R}^+$ are functions satisfying the following: $f$ and $\mu$ are continuous. $f$ is ...
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3answers
28 views

Find the area of the region described by $|5x|+|6y| \le 30 $

Find the area of the region described by $|5x|+|6y| \le 30 $ (where $|z|$ denotes the absolute value of $z$). My effort Imagining a number line and interpreting the problem as the request to ...
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1answer
24 views

Is torus w. disc removed homotopic to klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know f and g are homotopic if they represent: ...
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2answers
41 views

Find $f$, when $f(1)=f(2)=1$ and $f'(x)\leq (x^2-x-1) e^x, \forall x\in [1,2]$

Let $f:[1,2]\to\mathbb{R}$ be a differentiable function such that $f(1)=f(2)=1$ and $f'(x)\leq (x^2-x-1) e^x, \forall x\in [1,2]$. Find $f(x)$. I am pretty sure that from things I have tried ...
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0answers
23 views

Any mathematical formulation for this simple allocation problem?

General idea: Lets say there's a finite supply (S) of resources which should be allocated to two tasks based on their demands, D1 and D2. If Supply is larger than total demand (D), each task simply ...
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0answers
10 views

Determining function of Graph

Looking through some notes on numerical computation I came across the following graph: I know this is a longshot, but I'm not incredibly mathsy and would like to know what kind of a function this ...
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2answers
52 views

A weaker than “zero derivative” condition implies that a function is constant?

Let $f:(a,b)\to R$ be a continuous function such that $\limsup\limits_{n\to \infty}\frac{f(x_n)-f(x_0)}{|x_n-x_0|}\leq 0$ for every $x_0\in(a,b)$ and sequence $x_n$ converging to $x_0$ such that ...
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3answers
42 views

Taylor expansion of $\cos{x}$

I found a pdf file on the internet which gives you known expansions of Taylor's. There is something I cant understand : Why is the remainder of $\cos x$ is written like this? $$\frac{\cos ...
3
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3answers
32 views

Determine whether $e^x(1-e^x)\leqslant (1/4)$ for $x\lt 0$ is true or false?

The statement is $e^x(1-e^x)\leqslant (1/4)$ for $x\lt 0$ I think the above statement is true by calculating approximate value at some points. But how to prove it properly?
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1answer
19 views

Discrete Math Compositions

I am having trouble with these compositions. $$T = \{(a,a), (a,b), (b,c), (b,d), (c,d), (d,a), (d,b)\}$$ $$U = \{(a,a), (a,d), (b,c), (b,d), (c,a), (d,d)\}$$ I need to find $T \circ T$, $U \circ T$, ...
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1answer
52 views

Finding the domain of $\frac{1}{x}|x^2 - 1|$ [on hold]

What is the domain of this function $F(x)=\frac{1}{x}|x^2 - 1|$ Can someone please tell me how to find it ?
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2answers
22 views

Why coefficients have to be proportional for two quadratic functions to have the same roots?

We have the next two quadratic functions: $ ax^2 + bx + c = 0 $ $ mx^2 + nx + p = 0 $ If $ a/m = b/n = c/p $ then they have the same roots. What is the intuition behind this statement?
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31 views

Find the inverse of the following piecewise defined function

Find the inverse of $f$ if $f(x)=$ $$ \begin{cases} \sqrt{2-x}, &\text{for $x<0$}\\ 1-x^2, &\text{for $x \ge 0$} \\ \end{cases} $$ My effort For $y=\sqrt{2-x}$ ,we find ...
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0answers
16 views

Need function/formular for time manipulation book

this might be a rather unusual question, however: I'm working on a book idea and am looking for a mathematical function or some formular etc. Here is what I have in mind, so you might get a better ...
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2answers
62 views

Why is the function continuous at a point which gives the case 0/0?

I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = ...
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0answers
20 views

Calculus: Proving Continuous Function by Intermediate Value Theorem [duplicate]

Prove step by step: Let $f(x)$ be a continuous function from the closed interval $[a, b]$. Use the Intermediate Value Theorem to show that $f(x)$ has a fixed point, that is, there is a point $x \in ...
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2answers
28 views

What is the instantaneous rate of change in the real world?

I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place? In this moment I just know that it is named the ...
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0answers
39 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
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2answers
42 views

Uniqueness of log function with relaxed conditions?

Question If: $$f(a) + f(b) = f(ab)$$ $$ f(1) = 0 $$ $$ a<b \implies f(a) < f(b) \forall a,b \in N $$ where $N$ is the set of natural numbers. Prove or disprove $f$ must be the $\log$ ...
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45 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx$,is ...
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1answer
51 views

In which cases are $(f\circ g)(x) = (g\circ f)(x)$?

I have found three cases: 1) If $f$ and $g$ are the same function. 2) If $f$ and $g$ are mutually inverse. 3) If both are polynomials of degree $1$ Maybe there are more.
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0answers
31 views

how to get the solution?

I take the next exercise from Apostol's book. I thought for a while , but nothing. Then I read the solution. I try to think: how did he get it , but nothing. I want to know how to get to the solution. ...
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1answer
22 views

Function whose $n$-th derivative at $x=0$ is $n^3$ / evaluating the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$

I have troubles proving that the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$ represents the function $f(x)=e^x(x^3+3x^2+x)$. My idea was using the identity theorem for power series and the ...
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2answers
35 views

Finding ranks and nullities of linear maps

I am confused about ranks, nullities and bases of the kernel. From what I understand the rank is the dimension of a vector space generated by a matrix. How would I do the following examples? Find ...
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2answers
24 views

Vector-Valued Functions and Continuity

Why is it that when a vector-valued function $r(t)$ is continuous at some time $t$ then $\|r(t)\|$ is also continuous at that time $t$, but the converse is not true? That if $\|r(t)\|$ is continuous ...
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2answers
36 views

Show that an inverse of a bijective linear map is a linear map.

So I've got a bijection. It clearly has an inverse, but how exactly do I prove that the inverse is a linear map as well? enter image description here
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2answers
49 views
+200

Codomains and the definition of a function

A function $f$ is defined as a set of ordered pairs $(x, y)$ such that $(x,b), (x,c) \in f \Longrightarrow b=c$. Since $y$ is determined uniquely by $x$, it is customarily denoted $f(x)$. One ...
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2answers
49 views

Find the inverse $\dfrac{x}{\|x\|}$ in $\mathbb{R^2}$

I wish to find the inverse of $\dfrac{x}{\|x\|}$, where $x \in \mathbb{R}^2$ Let's do this. Let $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}}$$ $$y_2 = \dfrac{x_2}{\sqrt{x_1^2+x_2^2}}$$ Then $$y_1 = ...
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1answer
20 views

How can I show the points of continuity of the following function

How can I show the points of continuity of the following function $$f(x) = \begin{cases} 2x, & \text{if $x \in \Bbb Q$} \\[2ex] x+3, & \text{if $x \in \Bbb I$ } \end{cases}$$ I am having ...
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2answers
17 views

Neural Network Sigmoid Problem

I currently try to create a three layer neural network and I want to use the backpropagation. My training data is: Input : Output 0 | 1 : 0 1 | 1 : 2 In many ...
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0answers
22 views

Function Limit & Continuity [on hold]

What is Function Limit & Continuity? I'm a little bit silly.Is there anyone to explain those terms precisely? Thanks in Advance...
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2answers
60 views

Help with continuity [on hold]

Could you please clarify these questions to me. Find all the numbers for which the given function is discontinuous. $F(x)=[x-1]$ I think the solution is $\Bbb Z$ numbers right ? $F(x)= ...
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1answer
15 views

If $f(r,t)=g(r),\,\forall r,t$ would that make $f$ and $g$ constant functions?

I know that if $f(r)=g(t),\,\forall r,t$ then $f(r)=g(t)=constant$, but If $f(r,t)=g(r),\,\forall r,t$, where now $f$ depends on $t$ would that lead to the same conclusion i.e. ...
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0answers
37 views

Simplify $|a+sx|$ if $|x|=x$ and $|s|=1$. [on hold]

Simplify $|a+sx|$ if $|x|=x$ and $|s|=1$. Forgive me, this is probably really simple. Thanks. Edit: $a=v+sx$, where $v$ is a symmetric random variable with zero mean. So we can't say anything ...
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0answers
16 views

Is there a way to reverse engineer an already large number to make it smaller? [on hold]

Using the Ackermann function for example, it's quite easy to make massive numbers. My question is whether there's an existing algorithm that can take a large number and reverse engineer it to make an ...
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1answer
31 views

Let $S=[0,1) \cup [2,3]$ and $f:S \to \Bbb R$ be a strictly increasing map such that $f(S)$ is connected. Which of the following statements is true?

$f$ has exactly one discontinuity. $f$ has exactly two discontinuities. $f$ has infinitely many discontinuities. $f$ is continuous. I know theorems related to connectedness and ...
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2answers
43 views

Functions between metric spaces (and how they relate to closures of sets)

Let $(X,d)$ and $(Y , p)$ be metric spaces. Prove that if $f : X \to Y$ is continuous, then for any set $A\subset X$ with closure $\overline{A}$ we have $f(\overline{A})\subset \overline{ f(A) }$ ...
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0answers
96 views

How do I evaluate this special type of integral

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
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1answer
25 views

Is tangent monotonically increasing?

According to wolfram a function is monotonic if its derivative never changes sign, but the derivative doesn't have to be continuous. So I feel the answer is Yes, tangent is monotonically increasing. ...
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2answers
33 views

Prove that there is C such that $f(c)=0$

$$f\in C[a,b] $$If given that for every $ x\in[a,b]\text{ there exists } y\in[a,b]$ such that: $$|f(y)| \le \frac{1}{2}|f(x)| $$ Prove that there exists $ c\in[a,b]$ such that : $$f(c)=0$$ What ...
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1answer
41 views

Decomposition of function of bounded variation

Suppose we have $f:\mathbb{R} \rightarrow \mathbb{R}$ which is of bounded variation. I would like to show that it can be presented as a sum of left and right continuous functions of bounded variation. ...
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5answers
144 views

Why does $\DeclareMathOperator{arccot}{arccot}\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$ diverge?

Why does $$\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$$ diverge? In my textbook it says that from the positive side it's zero, and from the negative side it's $\pi$. However, when entering ...
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1answer
26 views

Bound the variation of this decreasing function

Let $f(x)$ be a decreasing function defined over the interval $[0,a]$, with $f(0)=b$. The first derivative of $f(x)$, which is negative, is such that $f^\prime(x) > g(x)$, or equivalently ...
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1answer
11 views

Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
0
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0answers
22 views

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well.

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well. Let $f,g \in C(\Omega)$. Since $f\ne g$, there is a $x_0 \in \Omega$ such that $f(x_0)\ne g(x_0)$. Hence ...
0
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1answer
29 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
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2answers
29 views

Asymptotes of $\arctan (2x)$

My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, ...