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

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9
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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
3answers
308 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
804 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
741 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
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4answers
544 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
536 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
10k 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
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4answers
702 views

Can the exponential function be reprsented as infinite product?

Is there any representation of the exponentil function as infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e. $$\mathrm ...
9
votes
4answers
251 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 ...
9
votes
2answers
496 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
378 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!
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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
748 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 ...
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votes
1answer
355 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
314 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
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
312 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
1answer
101 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
votes
1answer
81 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
158 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
243 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
229 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
999 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
8answers
525 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
6answers
980 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
4answers
720 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? ...
8
votes
3answers
618 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
2answers
925 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 ...
8
votes
4answers
524 views

Is $\tan\theta\cos\theta=\sin\theta$ an identity?

A friend of mine, who is a high school teacher, called me today and asked the question above in the title. In an abstract setting, this boils down to asking whether an expression like "$f=g$" is ...
8
votes
4answers
415 views

Can the Identity Map be a repeated composition one other function?

Consider the mapping $f:x\to\frac{1}{x}, (x\ne0)$. It is trivial to see that $f(f(x))=x$. My question is whether or not there exists a continuous map $g$ such that $g(g(g(x)))\equiv g^{3}(x)=x$? ...
8
votes
2answers
629 views

Is a function changed into another function by a change of variables?

If I have a function $ u(x,t) = p(x+ct) + q(x-ct) $ (which is the d'Alembert solution to the $1D$-wave equation), I can make the substitutions $$ \xi(x,t) = x + ct\\ \eta(x,t) = x - ct $$ So I am ...
8
votes
3answers
1k views

Proof of a simple property of real, constant functions.

I recently came across the following theorem: $$ \forall x_1, x_2 \in \mathbb{R},\textrm{function, } f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto y; \ |f(x_1) - f(x_2)| \leq (x_1-x_2)^2 \implies ...
8
votes
5answers
591 views

General Introduction to Functional and other Mathematic Notations

I've been a programmer for a good while now. Fairly experienced at a bit of math as far as coming up with algorithms and such but I am far far behind on understanding quite a deal of notation. Here ...
8
votes
5answers
543 views

Study continuity of this function

Hello im studying calculus at the university and I dont know how to solve the following exercise: Study the continuity of the next function: $$f(x,y) = \begin{cases} \frac{x^2-xy}{x+y}&\text{for } ...
8
votes
1answer
282 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
292 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
454 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.
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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
388 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 $x \in [0,1]$. ...
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votes
3answers
343 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
4answers
700 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
1answer
705 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 ...
8
votes
2answers
51k 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
276 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
914 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 ...