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

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9
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3answers
326 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
825 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
5answers
1k views

Is every injective function invertible?

Is every injective function invertible? How could I prove such thing? (Or is it just a necessary but not sufficient condition?) If $f:A\rightarrow B$ is injective then $f(a) = f(b) \Rightarrow a = b$ ...
9
votes
2answers
1k 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 ...
9
votes
2answers
1k 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
4answers
585 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
588 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
3answers
374 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?
9
votes
4answers
259 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
4answers
1k 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 ...
9
votes
2answers
509 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
4answers
192 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 ...
9
votes
3answers
6k 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 ...
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votes
4answers
1k 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 ...
9
votes
2answers
2k 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
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 ...
9
votes
2answers
158 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 ...
9
votes
1answer
11k views

“Well defined” function - What does it mean?

What does it mean for a function to be well-defined? I encountered with this term in an excersice asking to check if a linear transformation is well-defined.
9
votes
1answer
231 views

finding the value of $f(2001) $ if…

if $f (\frac{x}{y}) =\frac{f(x)}{y} $ and $f(2000)=1$ ; then what's the value of $f(2001)$. I tried hard but can't figured out anything. please help me, how can I proceed?
9
votes
1answer
363 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
1answer
766 views

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
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3answers
270 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
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3answers
323 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
198 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
83 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
2answers
309 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
143 views

Counting number of mathematical objects and structures

Regarding the numbers of certain mathematical objects and structures, especially sets, relations and functions, I've compiled a list of the counts from various sources: Partitions of a set with $k$ ...
9
votes
1answer
127 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 ...
9
votes
1answer
93 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
180 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
259 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 ...
9
votes
2answers
146 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.
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votes
0answers
372 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
6answers
6k views

Max Min of function less than Min max of function

I can't understand why max min of a function is less than equal to min max of that function i.e Why $$\underset{x}{\text{max}}\:\underset{y}{\text{min}} f(x,y) \leq ...
8
votes
8answers
293 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
6answers
1k 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
671 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
781 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
160 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.
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8answers
756 views

How to denote “powers” of a function?

I'm working with functions themselves, and I have learned that functional powers mean composition so: $f^3 = f \circ f \circ f$ But I'm looking for something that means $fff$. So $(fff)(x) = ...
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votes
4answers
591 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 ...
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votes
4answers
442 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
671 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
606 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 ...
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votes
2answers
7k views

Prove that the only eigenvalue of a nilpotent operator is 0?

I need to prove that if $\phi : V \rightarrow V$ is nilpotent, then its only eigenvalue is 0. I know how to prove that this for a nilpotent matrix, but I'm not sure in the case of an operator. How ...
8
votes
5answers
500 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
5answers
671 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
289 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
299 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 ...