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

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8
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5answers
470 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
4k 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
400 views

Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$

Suppose that $f(x)$ is differentiable on $[0,1]$ and $f(0) = f(1) = 0$. It is also known that $|f''(x)| \le A$ for every $x \in (0,1)$. Prove that $|f'(x)| \le A/2$ for every $x \in [0,1]$. ...
8
votes
4answers
680 views

Is the following a valid mathematical statement?

For all $f:\mathbb N\to\{1,2,3,\ldots,100\}$, If $f$ is a one to one correspondence, Then $f^{-1}(2)=3$ It seems as though this should not be a valid statement, since the implication fails to ...
8
votes
4answers
726 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 ...
8
votes
3answers
345 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 ...
8
votes
2answers
53k views

Types of polynomial functions. Quadratic, cubic, quartic, quintic, …,?

I would very much like to have a complete list of the types of polynomial functions. I know that theres: ...
8
votes
2answers
283 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
282 views

$f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ twice

Does there exist a continuous function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ exactly two times?
8
votes
4answers
951 views

Proof of linear independence of $e^{at}$

Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$, form a linearly independent set ...
8
votes
2answers
786 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
4answers
211 views

if the function $f(f(n))+f(n)=2n+2014$,find the $f$

let the function $f:N^{+}\to N^{+}$,and such $$f(f(n))+f(n)=2n+2014$$ Find the $f(n)$ My try: let $n=1$,then we have $$f(f(1))+f(1)=2016$$ let $f(1)=a$,then $$f(a)+a=2016$$ and let ...
8
votes
2answers
151 views

What is the meaning of $\mathbb{R}\setminus\{0\}$?

This is used in many posts related to functions and googling it doesn't help. What does this mean? $\mathbb{R}$ should stand for all Real numbers.
8
votes
1answer
667 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 $m|n$.
8
votes
7answers
240 views

Prove that $f(x)\equiv0$ on $\left[0,1\right]$

Let $f(x)$ differentiable on $\left[0,1\right]$ such that $f(0) = 0$. Also, assuming that $\forall x\in \left[0,1\right]:\left|f'(x)\right| \le \left|f(x)\right|$. Prove that $f(x)\equiv 0$ What I ...
8
votes
3answers
81 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
195 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
244 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
2k 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
189 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
117 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
2answers
418 views

if $|f(n+1)-f(n)|\leq 2001$, $|g(n+1)-g(n)|\leq 2001$, $|(fg)(n+1)-(fg)(n)|\leq 2001$ then $\min\{f(n),g(n)\}$ is bounded

The following question was proposed at MOP 2001 A function $f:\mathbb{N}\to\mathbb{N}$ is called cautious if $|f(n+1)-f(n)|\leq 2001$ for all $ n\in\mathbb{N} $. Suppose that $f,g,h$ are ...
8
votes
2answers
105 views

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large \frac{1}{z}$ by definition discontinuous at $0$?

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large{\frac{1}{z}}$ by definition discontinuous at $0$? Personally I would say: "no". In my view a function can only ...
8
votes
3answers
260 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
2answers
100 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
1answer
410 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
90 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
4answers
125 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
8
votes
3answers
182 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
237 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
143 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
1answer
243 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
81 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
99 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
287 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
217 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
351 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
96 views

How many expressions can be formed with two commutative and associative functions?

EDIT: I have posted a generalization of this question to MathOverflow here. Suppose we have two binary functions $f,g$ which are commutative and associative, i.e., satisfying $$ f(a,b) = f(b,a) ...
8
votes
1answer
57 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
101 views

Functional equation $f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}$

Problem: Find all continuous real-valued functions $f$ such that $$f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}.\tag{1}$$ Here $f$ is allowed to be defined only on a subset of $\mathbb{R}$. The only ...
8
votes
1answer
48 views

Find the function of separation between two functions

I seriously doubt that is what it is actually called, but I'm not very knowledgeable in this matter. Conceptually, what I am trying to do is calculate the function of a line/curve that shows the ...
8
votes
1answer
115 views

Exercise about continuous functions

Consider a continuous function $f \, : \, [0,1] \, \longrightarrow \, [0,+\infty)$ such that $f(0)=f(1)=0$ and : $\forall x \in (0,1), \; f(x) > 0$. I would like to prove that there exist ...
8
votes
2answers
125 views

Example of function with a hundred minima

Find a function $f\colon\mathbb{R}^2 \to\mathbb{R}$, $f \in C^{\infty}$, such that $\nabla f = 0$ for exactly $100$ points and in these points there are only local minima.
8
votes
0answers
246 views

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
8
votes
0answers
354 views

Compositional “square roots”

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
8
votes
0answers
348 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 ...
8
votes
1answer
214 views

When does the “Zetor function” converge?

Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. ...
8
votes
2answers
217 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 $x\in(a,b)$: ...
7
votes
6answers
983 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