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

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7
votes
3answers
110 views

How can we prove this function must be linear?

Let $f:[a,b]\to\mathbb{R}$ be continuous. Suppose for any sequence $(r_n)_{n=0}^{\infty}$ with $\lim_{n\to\infty}r_n=0$, and any $x\in(a,b)$: ...
1
vote
2answers
436 views

Prove $f(x,y) = xy/(x^2 + y^2)$ is continuous everywhere except $(0,0).$

I'd just like to ask you if my proof here is valid. I'll provide you with the method I used and if it seems ok let me know! If not, explanations would be helpful! My main approach to this question ...
-1
votes
3answers
30 views

Optimization with contraint

Given the value K with constraint x+y = K, what can be the maximum value of x*y be? How did they derive this answer? It is equivalent to finding the maximum value of x*(K-x), which will happen when x ...
0
votes
1answer
49 views

Help to clarify inductive step for proof of $\mathbb{N_m} \rightarrow \mathbb{N}_n\Rightarrow m\le n$

The statement to prove is: If there exists an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$ The solution says to prove by induction on $n$. I just need help on the inductive ...
0
votes
1answer
36 views

Figuring the function $f(x)$ from given information

Here is the given information in my question, So, what my question inform is that there is a cubic polynomial function (i.e $f(x)$) which has local maxima at $x=-1$. While that for $f'(x)$, it's ...
0
votes
2answers
20 views

Relations on set A and conditions of existence of descending chains

I am reading through the Elements of Set theory by Herbert Enderton and even though I have passed this exercise long ago,the more I look at my solution the less I believe it. Problem goes like this: ...
0
votes
0answers
33 views

Iterative function eventually reaching identity

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that ...
0
votes
1answer
30 views

how to find uniform continuity

I have some questions on continuity. What is the difference between continuous and uniformly continuous function? Please explain with this question. Find $f(x)=x^2$ is uniformly continous on ...
5
votes
2answers
195 views

Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: $g(1) = \alpha;$ $g(2n+j) = 3g(n) + γn + β_j$ : j = 0,1 and n >= 1 I have assumed the closed form to ...
1
vote
1answer
43 views

Alternative function definitions

If you go to the wikipedia page on the sine function or the log function you'll find a number of different definitions of these functions. I know that what defines a function are it's values, for ...
1
vote
0answers
29 views

Question about writing a proof with continuous functions [duplicate]

How would I write a proof for this example? We know that all polynomial functions on the reals are continuous by using the sequential definition of continuity. In particular, we know that the ...
-1
votes
0answers
24 views

Need some help with a function on a graph.

I need to make a function that starts pretty fast going up but then slows down but still goes up. Thanks in advance.
2
votes
1answer
50 views

Proof that there is a bijection, if there are injective maps in both directions

Let $A$ and $B$ be two sets. Let $f:A\to B$ be injective such that $Im(f) \subsetneq B$. Let $g:B\to A$ be injective such that $Im(g) \subsetneq A$. Obviously $A$ and $B$ are not finite sets. Can ...
1
vote
1answer
21 views

Convergent Sequence of Analytic Functions, Do Their Derivatives Converge?

If you have a sequence of real analytic functions that converge on every compact subset in your domain, do their derivatives necessarily converge to the derivatives of the function that they converge ...
8
votes
1answer
615 views

What is meant by “m|n”? Two letters separated by a vertical bar (|)

I am new to this subject, and not not sure what "|" symbol means on this statement. Let $R_2 \subset\Bbb N \times\Bbb N$ be defined by $(m, n) \in R_2$ if and only if $m|n$.
0
votes
1answer
44 views

How to prove that a given map is an injection?

Let $g:\mathbb{N_{m_1-1}}\rightarrow \mathbb{N}_{m_1}$, where: $$g(i) = \left\{ \begin{align} i & \text {, for } i<i_0 \\ i+1 & \text{, for } i \ge i_0 \end{align}\right.$$ and $i_0 ...
1
vote
2answers
99 views

Epsilon-Delta continuity definition for straight lines parallel to axes

I am taking a course on real analysis online and I encountered the $\epsilon-\delta$ definition for a function to be continuous. But I wonder if I can apply it to functions which are straight lines ...
8
votes
3answers
207 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
2
votes
1answer
22 views

Surjectiveness of standard-normal c.d.f. [closed]

Let $\phi:\mathbb R \to (0,1)$ be a function defined as $\phi(y)=\int_{-\infty}^y\dfrac{1}{\sqrt{2\pi}}e^{-\dfrac {x^2}{2}}dx , \forall y\in \mathbb R$ , then is it true that $\phi$ is surjective ? If ...
0
votes
1answer
22 views

An example of a twice continuously differentiable and bounded function.

Find an example of a twice continuously differentiable and bounded function $f:\Bbb R \rightarrow\Bbb R$ such that $\lim\limits_{x \rightarrow \infty} f(x)$ exists, but $\lim\limits_{x\rightarrow ...
0
votes
1answer
21 views

composing one function as a function of another function

I have two functions: $f_1=\sum_i x_iy_i$ and $f_2=\sum_i x_iy_i^2$, where $x_i$s and $y_i$s are positive and smaller than $1$. I want to write one of them as a function of the other (for example ...
1
vote
0answers
33 views

Why is there information loss here?

I posted this a while ago, but realized that actually this works, but produces information loss. Here is the process: Let's try to reduce a number $x$. We will be using base $\beta$ for this process. ...
2
votes
1answer
32 views

Composition and Limits

Suppose that $f$ is a continuously differentiable function with $\lim_{x \rightarrow \infty} f(x)=k$ and $g$ is a Lipschitz continuous function. Prove that $\lim_{x \rightarrow \infty} ...
1
vote
2answers
39 views

Fixed points of a certain type of functions with intermediate value property

Let $f: \mathbb R\to \mathbb R$ be a function, having intermediate value property, such that $f(f(x))=x , \forall x \in \mathbb R$, then is it true that either the set of fixed points of $f$ is ...
2
votes
3answers
34 views

How do I find the domain of this function

I would like to know which operations i have to do to get the domain of this function: $$y=\sqrt{\frac{1}{x}-1}$$ I have researched and the solution of the inequality $\frac{1}{x}-1 \geq 0$ is ...
0
votes
3answers
65 views

What would make a function reflexive, transitive, and/or symmetric?

A binary relation $R$ is a subset of the Cartesian product between two sets $X$ and $Y$, containing a set of ordered pairs $\{(x,y) : x \in X, y \in Y\}$. $R$ is a function if each element of $X$ is ...
0
votes
0answers
16 views

what are some good ways to define the errors between two functions?

I have two functions. One is the original function (that contains 4 variables), and the second one is the approximation to the first one (also contains 4 variables). The question is, if I want to ...
1
vote
1answer
10 views

Divisibilty of a functional equation

I found this question in a mathematical problems book: Let $f(x)$ is a polynomial such that $f(x^n)$ is divisible by $x-1$. Prove that $f(x^n)$ is divisible by $x^n-1.$ Can anybody help me?
0
votes
0answers
23 views

Can anyone give the equation of the inverse of radial projection from a tetrahedron to sphere?

$(x,y,z) \mapsto \bigg(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}} \bigg)$ This is the equation of the radial projection. I need the inverse of this ...
0
votes
0answers
18 views

Is the following subset a vector subspace of F(R,R). How to prove that it is a subspace.

Is the following subset a vector subspace of F(R, R)? The set of functions f : R → R such that f(x + 2) = f(x) for all x ∈ R I know this is obviously a subspace, but the mark scheme didn't ...
0
votes
1answer
20 views

Integral of a composition of piecewise linear function with polynomial

Fix a number $k > 0$ and let $$T(x) = \begin{cases} k &: x \geq k\\ x &: |x| < k\\ -k &: x \leq -k \end{cases}. $$ Define $S(s) = \int_0^s T(|x|^{m-1}x)\;dx.$ I want to show that ...
2
votes
0answers
17 views

Boundedness of a certain function defined on a closed bounded real interval

Let $I:=[a,b]$ be a closed bounded real interval , $f: I \to \mathbb R$ be a function such that for every $x \in I$ , $\exists \delta_x>0$ such that $f(x)$ is bounded $ \forall x \in ...
2
votes
2answers
74 views

Proof for Surjections

I'm reading through Basic Algebra I (which I enjoy so far. Thoughts on this for self-studying?) and am having a difficult time proving surjection. I believe I understand the concept, but when it comes ...
-1
votes
2answers
88 views

Showing that $A\rightarrowtail A \times \{x\}$ is a bijection

$A\rightarrowtail A \times \{x\}$ where $A$ is any set and $\{x\}$ is an arbitrary one-object set. How would I show the following is a bijection ( one to one and onto)? I know if I turn it into a ...
0
votes
0answers
38 views

L'Hospital's rule for higher derivatives

Let $u,v \in C^\infty(\mathbb{R})$, where $u(0) = 0$ and $v(0) = 0$ and $v'(0) \not= 0$. Then, one can define a function $f \in C^\infty(\mathbb{R}\setminus\{0\})$ by $f := u/v$. L'Hospital allows ...
0
votes
1answer
41 views

How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
0
votes
3answers
106 views

What is the $\lor$ symbol?

In researching the consensus algorithm, I came upon the consensus theorem: How does the $\lor$ symbol function?
-2
votes
1answer
25 views

parameterize the following functions [closed]

Help determining the parameterized solution of the following functions $$a) { \left( x-2 \right) }^{ 2 }+{ \left( y-1 \right) }^{ 2 }=4\quad if\quad 1\le y\le 3$$ $$b) \frac { { \left( x+3 \right) ...
-7
votes
0answers
21 views

How to check the availability of particular function [closed]

I want to check the availability of particular function that returns the value of results in binary like 0 or 1....How to check these function using mathematics...
-3
votes
0answers
29 views

Need a parabolic equation using two points and the slope at those points.

Can someone give me a function to solve any parabolic equation that has two known points with known slopes? Thanks much. Example: Point 1: (x1, y1), slope a Point 2: (x2, y2), slope b
1
vote
2answers
50 views

Problem Involving a Generalized Cartesian Product

Let $I$ be a set, and for each $i \in I$, let $U_i$ and $V_i$ be sets. Furthermore, suppose for each $i \in I$, there is a bijection $f_i:U_i \to V_i$. Prove that there is a bijection $g:\prod_{i \in ...
0
votes
1answer
33 views

How to deduce number of unordered distinct pairs using set operations and bijections

In (b) of the example, we are ask to calculate the number of ordered pairs of distinct positive integers. I like the first method's answer (using bijections, set operations) because it clearly shows ...
1
vote
0answers
31 views

Math Formula For A Loyalty System

Basically, I help someone manage a stream on twitch.tv. She uses a program that rewards the viewers with a virtual currency. For every 30 minutes they watch they get 1 point. Also, they get a 1 ...
0
votes
0answers
31 views

Exercise about vector functions of real variable

Parametrize in a clockwise direction by means of a continuous vector function into pieces, starting at the point $(1,1)$ the following curve. (Sorry for my bad English) $$c:\begin{cases} { \left( ...
1
vote
1answer
15 views

Convex and Concave Functions using Known Function Values

I am reading the classic Prospect Theory: An Analysis of Decision Under Risk (1979, Econometrica) by Kahneman and Tversky. I am not clear on something on page 278: ...
1
vote
2answers
599 views

Example of analytic piecewise-defined function

Does there exist an analytic everywhere, piecewise-defined function $f$ such that: $f(x) = g(x)$ for $x < k$ $f(x) = h(x)$ for $x>k$ $f(x) = r$ for $x=k$ With $g \ne h $ ($g$ not the same ...
2
votes
2answers
30 views

Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...
17
votes
3answers
1k views

I am searching for an unusual real-valued function.

It needs to fulfill following condition for all positive $x,y ∈ ℝ$: $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ any ideas?
2
votes
2answers
34 views

How to find the period of the sum of two trigonometric functions

I want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$
2
votes
4answers
115 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...