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

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8
votes
1answer
273 views

Function $f:\mathbb R^+\rightarrow \mathbb R^+$ that is eventually greater than $x^{x^{x^{…^{x^x}}}}$

For each $n$, define $f_n:\mathbb R^+\rightarrow \mathbb R^+$ by $f_n(x) = \underbrace{x^{x^{x^{...^{x^x}}}}}_n$ I want to find a function $f:\mathbb R^+\rightarrow \mathbb R^+$ such that for any ...
8
votes
4answers
480 views

A continuous function defined on an interval can have a mean value. What about a median?

A function can have an average value $$\frac{1}{b-a}\int_{a}^{b} f(x)dx$$ Can a continuous function have a median? How would that be computed?
8
votes
4answers
276 views

Is it known or where does this lead to?

I am eleventh class student, recently I started learning calculus. I was experimenting on various things, and found a new thing. It is as follows. Let us consider a function $f(x)$which is ...
8
votes
5answers
424 views

Does there exist a function $f:[0,1] \to[0,1]$ such its graph is dense in $[0,1]\times[0,1]$?

Does there exist a function $f:[0,1]\to [0,1]$ such that the graph of $f$ is dense in $[0,1]\times[0,1]$? Not necessarily continuous.
8
votes
2answers
3k views

What is a function to represent a diagonal sine wave?

I need to be able to plot pixels in this pattern. To me, it looks like a sine wave pattern that is both diagonal and convergent. What would a function for that look like? Thanks.
8
votes
2answers
90 views

$f: \mathbb N \to \mathbb N $ is injective $ \implies $ $\lim\Big(f(n)\Big)=+ \infty $?

If $f: \mathbb N \to \mathbb N $ is an injective function , then for the sequence $\Big(f(n)\Big)$ do we have $\lim\Big(f(n)\Big)=+ \infty $ ?
8
votes
2answers
271 views

Odd $C^\infty$ function

Suppose that $$f\in C^\infty (\mathbb{R})$$ and $f$ is an odd function. ($f(x)=-f(-x)$) What can we say about the zero at zero? Does $f$ have to be of the form $x g(x)$ for some $g\in C^\infty ...
8
votes
3answers
521 views

Linear independence of functions

I want to determine whether 3 functions are linearly independent: \begin{align*} x_1(t) = 3 \\ x_2(t) = 3\sin^2(t) \\ x_3(t) = 4\cos^2(t) \end{align*} Definition of Linear Independence: $c_1x_1 + ...
8
votes
2answers
646 views

On sort-of-linear functions

Background A function $ f: \mathbb{R}^n \rightarrow \mathbb{R} \ $ is linear if it satisfies $$ (1)\;\; f(x+y) = f(x) + f(y) \ , \ and $$ $$ (2)\;\; f(\alpha x) = \alpha f(x) $$ for all $ x,y \in ...
8
votes
3answers
80 views

$(\Bbb R \to \Bbb R : x\mapsto x^2)\equiv(\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2) \not\equiv (\Bbb C \to \Bbb C:x\mapsto x^2)$

Consider the following functions: $f:\Bbb R \to \Bbb R : x\mapsto x^2$ $g:\Bbb R \to \Bbb{R}_{\geq 0} : x \mapsto x^2$ $h:\Bbb C \to \Bbb C:x\mapsto x^2$ I'm quite sure that $h$ is not equal to $f$ ...
8
votes
3answers
177 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
8
votes
3answers
236 views

is the following a decreasing function?

I am stuck on figuring out why the following function is a decreasing function when I read a paper. The function is following $$f(x)=-\frac{1}{x}\log[{pe^{-ax}+(1-p)e^{-bx}}]$$ where $a$ and $b$ are ...
8
votes
3answers
1k views

chain rule using tree diagram, why does it work?

In multivariable calculus, I was taught to compute the chain rule by drawing a "tree diagram" (a directed acyclic graph) representing the dependence of one variable on the others. I now want to ...
8
votes
2answers
183 views

Continuous $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the continuous functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
8
votes
1answer
111 views

Find $f$ such as $f(x) = \sum_{n=1}^\infty \frac{f(x^n)}{2^n}$

Find $f \in C^0([0,1] , \mathbb{R})$ such as $$f(x) = \sum_{n=1}^\infty \frac{f(x^n)}{2^n}$$ My try : Constant functions work fine. We can notice : $$f(x) = \frac{f(x)}{2}+\sum_{n=2}^\infty ...
8
votes
3answers
216 views

Regarding the notation $f: a \mapsto b$

While I have come to understand that $f:a\mapsto b$ means that for input from the set $a$, the function will return a value from the set $b$, I am curious as to how far one may "drag" this notation. ...
8
votes
3answers
226 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 ...
8
votes
1answer
400 views

To get addition formula of $\tan (x)$ via analytic methods

Assume that we only know $\tan (0)=0$ and also given the relation $\tan'(x)=1+\tan^2(x)$ about $\tan (x)$ and we do not know other $\tan (x)$ relations of trigonometry. How can I get the additon ...
8
votes
1answer
85 views

Are there associative bijections $f\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$?

This is my first question and I hope this question is not too brief to be acceptable: There are well-known bijections $\mathbb{N \times N \to N}$ and well-known associative compositions on countably ...
8
votes
2answers
80 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
8
votes
3answers
176 views

How do I read this question? (subject: bijections)

Introduction In Basic Algebra I, I am struggling with fully understanding the following exercise: Show that $S\overset{\alpha}{\to}T$ is injective if and only if there is a map ...
8
votes
1answer
203 views

If the set of values , for which a function has positive derivative , is dense then is the function increasing?

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $A:=${ $x \in \mathbb R :f'(x)>0$ } is dense in $\mathbb R$ , then is it true that $f$ is an increasing function ? What ...
8
votes
1answer
118 views

If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$

Question Let the function $f:\mathbb R\to\mathbb R$,and such $$f(xf(y)+x)=xy+f(x)$$ Find all $f(x)$ Let $x=1,y=1$,then $$f(f(1)+1)=1+f(1)$$ let $f(1)=t$,then $$f(t+1)=1+t$$ So I guess ...
8
votes
3answers
3k views

Change of order of double limit of function sequence

The more general quesion is under what conditions the folloing equality will hold (all functions are real valued): $$\lim_{x \rightarrow a} \ \lim_{j \rightarrow \infty} f_j(x) = \lim_{j \rightarrow ...
8
votes
2answers
300 views

$ \sum\limits_{i=1}^{p-1} \Bigl( \Bigl\lfloor{\frac{2i^{2}}{p}\Bigr\rfloor}-2\Bigl\lfloor{\frac{i^{2}}{p}\Bigr\rfloor}\Bigr)= \frac{p-1}{2}$

I was working out some problems. This is giving me trouble. If $p$ is a prime number of the form $4n+1$ then how do i show that: $$ \sum\limits_{i=1}^{p-1} \Biggl( ...
8
votes
1answer
228 views

Prove that : $|f(b)-f(a)|\geqslant (b-a) \sqrt{f'(a) f'(b)}$ with $(a,b) \in \mathbb{R}^{2}$

Let $(a,b) \in \mathbb{R}^{2}$ such that $a<b$ and $f\in C^2([a,b],\mathbb{R})$ such that $f'\neq 0$ and $f''/f'$ is decreasing. Prove that : $$|f(b)-f(a)|\geqslant (b-a) ...
8
votes
1answer
62 views

Is every real valued function on an interval a sum of two functions with Intermediate Value Property?

If $I$ is an interval of real numbers , then is it true that any function $f:I \to \mathbb R$ can be written as $f=f_1+f_2$ , where $f_1 , f_2 : I \to \mathbb R$ have the Intermediate value property? ...
8
votes
1answer
91 views

A question of rationality

This problem was asked to me by a friend and I simply have no idea about it. So I have not progressed a single bit. The problem is this: If $f :\mathbb{R}\to \mathbb{R}$ is an infinitely ...
8
votes
4answers
180 views

Why isn't $y=(x^6)^{1/3}$ a polynomial function?

I've been told that $y=(x^6)^{1/3}$ isn't a polynomial function because of the radical but I believe that the equation could be simplified to $y=x^2$ which fits the definition of a polynomial ...
8
votes
1answer
201 views

Function mapping challange

For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$ and $f(k) = f(f(k + 1))$. This is ...
8
votes
2answers
346 views

Prove the following property of $f(x)$?

Let $$f(x)=|a_1\sin(x)+a_2\sin(2x)+a_3\sin(3x)+...+a_n\sin(nx)|.$$ Given that $f(x)$ is less than or equal to $|\sin(x)|$ for all $x$, prove that $|a_1+a_2+a_3+....|$ is less than or equal to ...
8
votes
1answer
53 views

About continuity

one question is disturbing me : let f and g two continuous (real-valued) functions on the unit interval [0,1] with the property that $[f(x)-f(y)][g(x)-g(y)]=0 ,\forall x,y \in [0,1]$. To my ...
8
votes
0answers
313 views

Multiplicative Identity analog for absolute value

Is there a standard name for the function: $$ f(x) = \begin{cases} x & \text{if |x|≤1;}\\ 1/x & \text{if |x|>1;}\\ \end{cases} $$ And is there a potential application of this ...
7
votes
5answers
805 views

Do we have always $f(A \cap B) = f(A) \cap f(B)$?

Suppose $A$ and $B$ are subsets of a topological space and $f$ is any function from $X$ to another topological space $Y$. Do we have always $f(A \cap B) = f(A) \cap f(B)$? Thanks in advance
7
votes
6answers
2k views

Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective

Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective? I tried the following: $$f:\mathbb{R}\rightarrow ...
7
votes
6answers
1k views

What do I not understand about one-to-one functions?

Firstly, a definition: Definition 1: A function $\phi : X \rightarrow Y$ is one-to-one if $\phi(x_1) = \phi(x_2)$ only when $x_1 = x_2$. Now the question: Students often misunderstand the ...
7
votes
7answers
376 views

Function satisfying $x = f(f(x))$ and $x \not= f(x)$

Is there a function that would satisfy the following conditions?: $\forall x \in X, x = f(f(x))$ and $x \not= f(x)$, where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in ...
7
votes
3answers
471 views

If $f \circ g = f$, prove that $f$ is a constant function.

Suppose $A$ is a nonempty set and $f: A \rightarrow A$ and for all $g:A \rightarrow A,$ $f \circ g = f$. Prove that $f$ is a constant function. This result seems obvious, but I can't seem to find ...
7
votes
2answers
747 views

Why use the derivative and not the symmetric derivative?

The symmetric derivative is always equal to the regular derivative when it exists, and still isn't defined for jump discontinuities. From what I can tell the only differences are that a symmetric ...
7
votes
2answers
321 views

Must $g$ be the identity if $f = g \circ f$?

I am finding it hard to solve the following problem. Let $A$ is a set and $f : A \rightarrow A$ and $g : A \rightarrow A$. If $f = g \circ f$, must $g$ be an identity function always? Will there be ...
7
votes
9answers
393 views

Nonpiecewise Function Defined at a Point but Not Continuous There

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...
7
votes
6answers
213 views

Representing the function $\mathbb Z_9\to\mathbb Z_9$, $f(0) = 1$, $f(1) = \ldots = f(8) = 0$ as a polynomial in $\mathbb Z_9[x]$

Let $\mathbb Z_9=\left\{0,1,2,3,4,5,6,7,8\right\}$ be the set of integers modulo 9 and $f:\mathbb Z_9 \rightarrow \mathbb Z_9$ be a function. Assume $f(0)=1$, $f(1)=f(2)=...=f(8)=0$. What is the ...
7
votes
2answers
241 views

Does $F(t)F'(t) \le 0$ imply that $F$ does not change signs?

The question is in the title, really. More precisely, suppose $F:[0,1] \to \mathbb{R}$ is continuously differentiable and satisfies (i) $F(t)F'(t) \le 0$ for all $t \in [0,1]$ (ii) $F(0) = 1$ ...
7
votes
4answers
651 views

Solving the functional equation $f(x+1) - f(x-1) = g(x)$

Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies $$ f(x+1) - f(x-1) = g(x) $$
7
votes
3answers
177 views

Is every function $f : \mathbb N \to \mathbb N$ a composition $f = g\circ g$?

True or wrong: For every function $f: \mathbb N \rightarrow \mathbb N$ there is a function $g: \mathbb N \rightarrow \mathbb N$ with $f=g \circ g$.
7
votes
4answers
439 views

Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?

Is this proof correct? To prove $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$. Let any number $y\in F(A)\cap F(B)$. We want to show $y\in F(A\cap B).$ Therefore, $y\in F(A)$ and ...
7
votes
2answers
238 views

Is there a non-constant smooth function mapping $\mathbb{R}$ into $\mathbb{Q}$

I cannot think of a non-constant smooth function which maps all real numbers into rational numbers. Can anyone give a simple example ? The simpler, the better !
7
votes
6answers
385 views

Rigorous Definition of “Function of”

When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a ...
7
votes
4answers
329 views

The leap to infinite dimensions

Extending this question, page 447 of Gilbert Strang's Algebra book says What does it mean for a vector to have infinitely many components? There are two different answers, both good: 1) The ...
7
votes
4answers
174 views

A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...