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

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10
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5answers
8k views

Calculating the total number of surjective functions

It is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ...
10
votes
1answer
219 views

Solving a special Quartic Equation.

Solve for $x$ $$(x^2-4)(x^2-2x)=2$$ I have tried the Rational Root Theorem and found that there are no rational roots. Further, the polynomial $p(x)=(x^2-4)(x^2-2x)-2$ is irreducible since ...
10
votes
3answers
861 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. ...
10
votes
1answer
882 views

Show f is uniformly continuous on $(a,b)$ if it is continuous and $\lim\limits_{x\to a^+}f(x)$ and $\lim\limits_{x\to b^-}f(x)$ exist

Let $f:(a,b)\to\mathbb{R}$ be continuous at all $x\in(a,b)$. If $\lim\limits_{x\to b^-}f(x)$ and $\lim\limits_{x\to a^+}f(x)$ exist in $\mathbb R$, how can we prove that $f$ is uniformly continuous on ...
10
votes
2answers
919 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 ...
10
votes
1answer
457 views

A question from the dreams realm

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a function (not necessarily continuous). Let $\phi_0(x)=\phi(x)$ and $\forall k\in\mathbb{N},\phi_{k+1}(x)=\phi(x\cdot\phi_k(x))$. 1. Let ...
10
votes
3answers
853 views

Infinitely differentiable function with given zero set?

For each closed set $A\subseteq\mathbb{R}$, is it possible to construct a real continuous function $f$ such that the zero set, $f^{-1}(0)$, of $f$ is precisely $A$, and $f$ is infinitely ...
10
votes
1answer
154 views

If $f:X \to X$ is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous?

If $f:X \to X$ (codomain and domain have the same topology) is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous? Note that the orbit being finite and $f$ being a ...
10
votes
2answers
296 views

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
10
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1answer
65 views

Simple true/false statement about function composition

Given the functions $f,g$ from $\mathbb{R}$ to $\mathbb{R}$ is it true that If $f \circ g$ is strictly increasing and $f$ is injective then $g$ is monotonic I believe this is false but I can't ...
10
votes
1answer
273 views

Approximating the area below average of a concave function

Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define \begin{align*} F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\} ...
9
votes
8answers
1k views

“Integer average” of two integer numbers

Suppose two arbitrary integer numbers $a$ and $b$. I'm looking for some function $f(a,b)$ with the following properties: $f(a,b)\in\mathbb{Z}$. $f(a,a)=a$. $f(a,b)=f(b,a)$. $\min\{a,b\}< ...
9
votes
5answers
1k views

Is it possible to combine two integers in such a way that you can always take them apart later?

Given two integers $n$ and $m$ (assuming for both $0 < n < 1000000$) is there some function $f$ so that if I know $f(n, m) = x$ I can determine $n$ and $m$, given $x$? Order is important, so ...
9
votes
4answers
732 views

If a function can only be defined implicitly does it have to be multivalued?

What is the general reason for functions which can only be defined implicitly? Is this because they are multivalued (in which case they aren't strictly functions at all)? Is there a proof? ...
9
votes
3answers
778 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
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votes
3answers
313 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...
9
votes
5answers
813 views

Why do we limit the definition of a function? [duplicate]

Why do we limit the definition of a function to only one y per x? For example, the square root function. We only allow the principal square root of a number, rather than, say, the square root of 9 ...
9
votes
2answers
791 views

Draw graph of $\frac{1}{f(x)}$ from graph of $f(x)$

If I know the graph of $f(x)$, how do I draw the graph of $\frac{1}{f(x)}$?
9
votes
6answers
1k views

How is the codomain for a function defined?

Or, in other words, why aren't all functions surjective? Isn't any subset of the codomain which isn't part of the image rather arbitrary?
9
votes
4answers
554 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?
9
votes
3answers
545 views

For how many functions $f$ is $f(x)^{2}=x^{2}$?

How many functions $f$ are there that satisfy $f(x)^{2}=x^{2}$ for all $x$? My text (Spivak's Calculus; chapter 7 problem 7) asks this question for continuous $f$, for which the answer is, of course ...
9
votes
4answers
254 views

injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$

Today a friend of mine told me a nice fact, but we couldn't prove it. The fact is that there is an injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ defined by the fomula $(m,n)\mapsto ...
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5answers
11k views

Number of onto functions

What are the number of onto functions from a set $\Bbb A $ containing m elements to a set $\Bbb B$ containing n elements. I ...
9
votes
1answer
724 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 ...
9
votes
2answers
499 views

Why doesn't this converge?

Why doesn't $$\int_{-1}^1 \frac{1}{x}~\mathrm{d}x$$ converge? I mean you would think that because of symmetry the area from the negative side and positive side cancel out, resulting in the integral ...
9
votes
1answer
395 views

What does it mean to extend a function?

What does it mean to extend a function? Can someone please give an example? Thanks in advance!
9
votes
2answers
1k views

Why are even/odd functions called even/odd?

Bit of a silly question, someone told me that the reason even functions are called 'even' and odd functions are called 'odd' is that all (single-variable) monomials with even powers are even functions ...
9
votes
2answers
768 views

Finite groups of functions under function composition

Over the years I have done many questions along the lines of the following: "Given functions $\phi, \theta$ (usually defined on $\mathbb{R}$ or $\mathbb{C}$, or a suitable subset of $\mathbb{R}$ or ...
9
votes
1answer
358 views

Proving the existence of a point with a certain property for a continuous function

Let $f:[0,1]\to\mathbb{R}$ a continuous function and $\int_0^1xf(x)dx=0$. Show that there exists a point $c\in(0,1)$ so that $f(c)=(\int_c^1f(x)dx)^2$. As a potential solution, I tried assuming that ...
9
votes
3answers
268 views

Why is this proof of $\mathbb{N}\times\mathbb{N}$ being countable not formal?

My copy of Introduction to Real Analysis: Bartle and Sherbert gives: Theorem: The set $\mathbb{N}\times\mathbb{N}$ is countable. Informal Proof: Recall that $\mathbb{N}\times\mathbb{N}$ ...
9
votes
3answers
319 views

Solve the functional equation $f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)$ with $f : [0,\infty) \to \mathbb R$ continuous

Solve the functional equation $$f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)\qquad \forall x\geq 0$$ with $f : [0,\infty) \to \mathbb R$ continuous. I can't manage to get this one ...
9
votes
1answer
192 views

Fourier Series involving the Jacobi Symbol

We know that the Fourier Series $$s(x)=\sum_{k\neq0}\frac{1}{k}\exp\left(2\pi ik x\right)$$ corresponds to the sawtooth function, $s(x)=\left\{x\right\} -\frac{1}{2}$. Suppose that ...
9
votes
1answer
58 views

If a set $S$ has a choice function, does $\bigcup S$ have one too?

I have an exercise in a book that asserts that if a set $S$ has a choice function on it, then so does the union of all its elements $\bigcup S$ (without assuming the axiom of choice). I, however, have ...
9
votes
3answers
14k views

Proving a function is onto and one to one

I'm reading up on how to prove if a function (represented by a formula) is one-to-one or onto, and I'm having some trouble understanding. To prove if a function is one-to-one, it says that I have to ...
9
votes
2answers
203 views

Is there a function such that $f' = f\circ f$?

Is there a function $f:\mathbb{R}\rightarrow (0,\infty)$, such that $f' = f\circ f$? Apparently, I should assume by contradiction there is, and then it should imply that $f$ is increasing but I ...
9
votes
2answers
341 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 ...
9
votes
2answers
308 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( ...
9
votes
3answers
244 views

Quadratic Equation Recurrence?

So I was playing around with the following: Let $f_0(x) = x^2 - bx + c$. If $f_n(x)$ has roots $p$ and $q$ with $p > q$, then let $f_{n+1}(x) = x^2 - px + q$. The recurrence relation is rather ...
9
votes
1answer
102 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 ...
9
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1answer
115 views

Non-computable function having computable values on a dense set of computable arguments

A rational complex number is a complex number whose both real and imaginary parts are rational numbers. Note that a rational complex number is a finitary object that can be an input or an output of an ...
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votes
2answers
134 views

Does equation $f(g(x))=a f(x)$ have a solution?

Suppose $g: [0,1] \to [0,1]$ is a strictly increasing, continuously differentiable function with $g(0)=0$ and $g(1) < 1$, and $a \in (0,1)$. Is there a function $f:[0,1] \to \mathbb{R}$ such that ...
9
votes
1answer
86 views

Interesting properties of the function $(a,b)\mapsto a/(a-b)$

Consider the extremely simple function $$f(a,b)=\frac a{a-b}.$$ This gives the coordinate where the line through $(0,a)$ and $(1,b)$ meets the $x$-axis. I noticed that the function $f$ has some ...
9
votes
2answers
169 views

What is the name of this property of a function

I'm trying to find the right vocab word to describe a concept: In computational geometry, there's a concept of a polygon "monotone" with respect to a line. Which means that the polygon intersects ...
9
votes
2answers
244 views

How do I find, algorithmically, which parts of a given function are interesting to graph?

I'm building a program that does 2D graphing, and was wondering: How can I determine the default zoom level and x/y extents to display on screen, in such a way as to maximise the 'interesting' parts ...
8
votes
8answers
270 views

An example of a mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that $\eta(x)=n$ has infinitely many solutions for each $n\in\mathbb{N}$

Suppose I have the mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that for each $n\in\mathbb{N}$ the equation $\eta(x)=n$ has infinitely many solutions. I saw this question which is basically the ...
8
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 ...
8
votes
6answers
999 views

what is this sort of function called?

I am doing an assignment but I do not know how to do this problem. I have the following: $$ f(x)= \begin{cases} 0 & \text{for $x<0$},\\ x & \text{for $x\geq 0$}. \end{cases} $$ We are ...
8
votes
8answers
556 views

Why is the Fibonacci ratio though a decreasing function, it is alternating and decreasing?

I tried to find the ratio of consecutive terms of the Fibonacci series and found that it is a decreasing function and it converges . I tried it though a small code piece in python so that I can have a ...
8
votes
3answers
638 views

Why is there no continuous log function on $\mathbb{C}\setminus\{0\}$?

Over the years, I've often heard that there is no logarithm function which is continuous on $\mathbb{C}\setminus\{0\}$. The usual explanation is usually some handwavey argument about following such a ...
8
votes
4answers
149 views

Is there an integral representation for $\frac{1}{n!}$?

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.