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

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12
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2answers
657 views

Isomorphic embedding of $L^{p}(\Omega)$ into $L^{p}(\Omega \times \Omega)$?

Let $(\Omega,\mu)$ be a finite measure space such that $\mu(\Omega)=1$. Suppose $1\leq p \leq \infty$. Let $\psi \colon L^p(\Omega) \to L^p(\Omega \times \Omega)$ be the map which maps $f$ onto the ...
12
votes
1answer
156 views

Let $(f(x))^2$ and $(f(x))^3$ are $C^{\infty}$. Prove or disprove that $f$ is $C^{\infty}$.

Suppose $f(x), -\infty < x < +\infty$, is a real valued function such that both $(f(x))^2$ and $(f(x))^3$ are $C^{\infty}$. Must $f$ be $C^{\infty}$? I had seen this exercise somewhere ...
12
votes
2answers
112 views

Function such that zeros$=$order of the derivative

Does there exist a function $f\in C^n(\mathbb{R},\mathbb{R})$ for $n\ge2$ such that $f^{(n)}$ has exactly $n$ zeros, $f^{(n-1)}$ has exactly $n-1$ zeros and so on ? Where $f^{(n)}$ is the nth ...
12
votes
3answers
5k 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 + ...
12
votes
3answers
321 views

$f(nx)\to 0$ as $n\to+\infty$

Let $f:\mathbb R^+\to\mathbb R^+$ be a continuous function, and let $I$ be a subset of $\mathbb R^+$ such that the following property holds: For any $x\in I$, $f(nx)\to 0$ as $n\to+\infty$. ...
12
votes
2answers
217 views

About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$

On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
12
votes
1answer
468 views

Countable-infinity-to-one function

Are there continuous functions $f:I\to I$ such that $f^{-1}(\{x\})$ is countably infinite for every $x$? Here, $I=[0,1]$. The question "Infinity-to-one function" answers is similar but without the ...
11
votes
5answers
2k views

Functions that are continuous only at two points?

I need to find a function $f:\mathbb{R}\to\mathbb{R}$ which is continuous only at two points, but discontinuous everywhere else. How on earth would I go about doing this? I can't think of any ...
11
votes
12answers
2k views

A function such that $f(f(n)) = -n$?

This question from somebody's job interview made me puzzled: Design a function f, such that: $f(f(n)) = -n$ , where n is a 32 bit signed integer; you can't use complex numbers arithmetic. If you ...
11
votes
2answers
2k views

Why the name 'FACTORIAL'?

Factorial is defined as $n! = n(n-1)(n-2)\cdots 1$ But why mathematicians named this thing as FACTORIAL? Has it got something to do with factors?
11
votes
5answers
634 views

(Non)Existence of limits

When we say that a limit of a function does not exist in $\mathbb{R}$ (or some metric space) does it make sense to say that it might exist somewhere else? [I am trying to think along lines of ...
11
votes
5answers
1k views

Exponential Function as an Infinite Product

Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e. $$\mathrm e^x=\prod_{n=0}^\...
11
votes
4answers
14k views

Prove $\sin x$ is uniformly continuous on $\mathbb R$

How do I prove $\sin x$ is uniformly continuous on $\mathbb R$ with delta and epsilon? I proved geometrically that $\sin x<x$ and thus, $$|f(x_1)-f(x_2)|=|\sin x_1 - \sin x_2|\le|\sin x_1|+|\sin ...
11
votes
4answers
389 views

$f(16x)=16f(x) $ and $ f$ is continuous

$f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(16x)=16f(x)$ for every real $x$. Should it be $f(x)=ax$? How can I prove that?
11
votes
2answers
3k views

Are there other kinds of bump functions than $e^\frac1{x^2-1}$?

I've only seen the bump function $e^\frac1{x^2-1}$ so far. Where could I find examples of functions $C^∞$ on $\mathbb{R}$ that are zero everywhere except on $(-1,1)$? Are there others that do not ...
11
votes
4answers
410 views

Find all functions f such that $f(f(x))=f(x)+x$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(f(x))=f(x)+x, \forall x\in\mathbb{R}$. Find all such functions $f$. Clearly, $f$ is an "one-to-one function". I have tried setting where $...
11
votes
4answers
3k views

When do two functions become equal?

When do two functions become equal? I have stumbled over this definition of equality of functions in elementary real analysis. Let $X$ and $Y$ be two sets. Let $f:X\rightarrow Y$ and $g:X\...
11
votes
2answers
314 views

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable? Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
11
votes
5answers
1k views

Can this function be rewritten to improve numerical stability?

I'm writing a program that needs to evaluate the function $$f(x) = \frac{1 - e^{-ux}}{u}$$ often with small values of $u$ (i.e. $u \ll x$). In the limit $u \to 0$ we have $f(x) = x$ using L'Hôpital's ...
11
votes
1answer
839 views

Who came up with the arrow notation $x \rightarrow y$?

I read that the arrow notation $x \rightarrow y$ was invented in the 20th century. Who introduced it? Each map needs both an explicit domain and an explicit codomain (not just a domain, as in ...
11
votes
3answers
192 views

How many different functions we have by only use of $\min$ and $\max$?

We can making many functions of three variable by only use and combining of $\min$ and $\max$ functions. But many of them are not different , like : $$\min(x,y,z)=\min(x,\min(y,z)),\quad\min(x,\max(x,...
11
votes
2answers
318 views

When is $f^{-1}=1/f\,$?

I seem to remember a nice article that I read many years ago (perhaps in the American Mathematical Monthly) which investigated the question, "Under what conditions on $f:\mathbb{C}\to\mathbb{C}$ is $f^...
11
votes
4answers
351 views

If $f(3x)=f(x)$ and $f$ is continuous, show that $f(x)$ is a constant function.

If $f(x)$ is a continuous function such that $f(3x)=f(x)$ and the domain of $f$ is all non-negative real numbers. Prove that $f$ is a constant function. What I did: $$f(3x)=f(x)=f\left(\frac{x}{3}\...
11
votes
2answers
223 views

Can every real function be represented as two shifted even functions?

I saw the theorem that every function can be represented as the sum of and even and odd function, and this made me wonder: can every function from the reals to the reals, defined on all the reals, be ...
11
votes
4answers
2k views

Difference between a function and a graph of a function?

Formally, I learned that a function $f: X \to Y$ is a subset $f \subset X \times Y$ subject to the condition that for every $x \in X$, there is exactly one $y \in Y$ such that $(x, y) \in f$. We write ...
11
votes
3answers
7k views

How $x^4$ is strictly convex function?

I hear that $f(x)=x^4$ is a strictly convex function $\forall x \in \Re$. However, strict convexity condition is that the second derivative should be positive $\forall x \in \Re$. For the mentioned ...
11
votes
1answer
175 views

Differentiation of a function $f:\mathbb{Q}\to \mathbb{Q}$(Rational Calculus)

Assume that $f:\mathbb{Q}\to \mathbb{Q}$ is given such that $\forall a\in \mathbb{Q}$ the following limit, exists \begin{equation} \lim_{x\to a} \frac{f(x)-f(a)}{x-a}\in \mathbb{R} \end{...
11
votes
5answers
4k views

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same?
11
votes
4answers
1k views

What was the notation for functions before Euler?

According to the Wikipedia article, [Euler] introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical ...
11
votes
2answers
88 views

Does there exist a function $g\in \mathbb{N}^\mathbb{N}$ s.t. $\{f\mid f\circ f=g\}$ is not empty and finite?

I'm struggling with this question and can't figure it out. The question was too long for the title so I will write it once more: Does there exist a function $g : \mathbb{N} \longrightarrow \mathbb{N}$...
11
votes
2answers
1k views

What exactly is the fixed field of the map $t\mapsto t+1$ in $k(t)$?

Suppose $k$ is a field, and $k(t)$ is the rational function field. If $f(t)=P(t)/Q(t)$ for some polynomials $P(t)$ and $Q(t)\neq 0$, then the map $t\mapsto t+1$ sends $f(t)$ to $f(t+1)$. So the ...
11
votes
2answers
234 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 ...
11
votes
2answers
542 views

Why are fractal curves nowhere differentiable?

I am a highschool student who stumbled upon fractals when doing a math project. In my research about fractals, I have found that they are nowhere differentiable. Can someone explain this in simple ...
11
votes
3answers
1k views

What happens to a function when it is undefined?

If I have the function $$f(x) = {x^2 - 2 \over x + \sqrt 2}$$ this is undefined for $x = -\sqrt 2$, am I correct? Since the denominator would be zero. But the numerator is a difference of ...
11
votes
3answers
174 views

Not every function on $[0,1]$ is a pointwise limit of continuous functions on $[0,1]$

How to show that not every $\mathbb{R}$-valued function on $[0,1]$ is a pointwise limit of continuous $\mathbb{R}$-valued functions on $[0,1]$? There is a theorem that states that the set of points ...
11
votes
1answer
414 views

Math Olympiad - pre-periodic function

Let $c \in \mathbb{Q}$, $f(x)=x^2+c$. Define $$f^{0}(x)=x, \ \ f^{n+1}(x)=f(f^{n}(x)), \ \forall n \in \mathbb{N}$$ We say that $x \in \mathbb{R}$ is pre-periodic if $\{f^{n}(x), n \in \mathbb{N}\}$ ...
11
votes
1answer
100 views

How to find the inverse of $f(x)=x+ \frac{x^{3}}{1+x^{2}}$?

I know that given, $f(x)=x+ \frac{x^{3}}{1+x^{2}}$ I should set $y=x+ \frac{x^{3}}{1+x^{2}}$ and solve in terms of $x$, then just swap the $x$'s and $y$'s. I know that, since the derivative is ...
11
votes
1answer
139 views

How do I construct a function $\operatorname{sog}$ such that $\operatorname{sog}\circ\operatorname{sog} = \log$?

Imagine a real-valued semilog function $\DeclareMathOperator{\sog}{sog}\sog$ with the property that $$\sog(\sog(x)) = \log(x)$$ for all real $x>0$. My questions: Does such a function ...
11
votes
1answer
126 views

$\max(a,b)=\frac{a+b+|a-b|}{2}$ generalization

I am aware of an occasionally handy identity: $$\max(a,b)=\frac{a+b+|a-b|}{2}$$ However, I have found I'm unable to come up with a nice similar form for $\max(a,b,c)$. Of course I could always use ...
11
votes
1answer
348 views

Additive functional inequality

The function $f:R_+\to R_+$ is continuously differentiable and increasing. Also, $f(0)=0$ and $f(\infty)=\infty$. Continuity and differentiability of higher orders can be assumed if necessary. ...
11
votes
1answer
1k views

Continuous and Open maps

I was reading through Munkres' Topology and in the section on Continuous Functions, these three statements came up: If a function is continuous, open, and bijective, it is a homeomorphism. If a ...
11
votes
1answer
189 views

If every point is a local maximum, is it a step function?

What are the functions $f:\mathbb R\to\mathbb R$ such that every point is a local maximum? Certainly, $f(x)=c$ works for every constant. So does $\lfloor x\rfloor$, as does $I_{\{0\}}(x)=\begin{cases}...
11
votes
3answers
141 views

If $B(x+y)-B(x)-B(y)\in\mathbb Z$ can we add an integer function to $B$ to make it additive?

Given a function $B:\mathbb R\to\mathbb R$ satisfying $B(x+y)-B(x)-B(y)\in\mathbb Z$ for all real numbers $x$ and $y$, is there a function $Z:\mathbb R\to\mathbb Z$ such that $B+Z$ is an additive ...
11
votes
0answers
396 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 ...
10
votes
3answers
1k views

A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
10
votes
8answers
358 views

Why is it important to have a discrepancy between image and codomain?

A function $f:\mathbb{R}\rightarrow \mathbb{R}$ given by $f(x)=x^2$ has $\mathbb{R}_{\geq0}$ as its image and $\mathbb{R}$ as its codomain. What's the need for this discrepancy? Why don't we just ...
10
votes
5answers
1k views

What does orthogonality mean in function space?

The functions $x$ and $x^2 - {1\over2}$ are orthogonal with respect to their inner product on the interval [0, 1]. However, when you graph the two functions, they do not look orthogonal at all. So ...
10
votes
3answers
4k views

When functions commute under composition

Today I was thinking about composition of functions. It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse. But then ...
10
votes
5answers
276 views

function identity: does $\frac{x^2-4}{x-2} = x+2$

Can you tell me how to resolve the (apparent) paradox that the function: $$f(x) = \frac{x^2-4}{x-2}$$ is identical to the function: $$g(x) = x+2$$ because by factoring the numerator: $$f(x) = \...
10
votes
4answers
175 views

Find the value of $ [1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$

Assume that [x] is the floor function. I am not able to find any patterns in the numbers obtained. Any suggestions? $$[1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$$