2
votes
1answer
33 views

What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
1
vote
1answer
30 views

Problem on function definition

I'm trying to solve this problem about functions: "Explain why $x^2-4=0$ is not a real function of real variable." I have yet solved many similar problems, but now i have a doubt; is my solution ...
4
votes
2answers
37 views

What is a transformation?

I am not a native English speaker and I have been pointed out that the word "transformation" as a synonym of "function" is grammatically incorrect. However, I even found a wikipedia and a mathworld ...
6
votes
3answers
131 views

Functions with different codomain the same according to my book?

My book gives the following definition: A function $f$ from $A$ to $B$ is defined as $f\subseteq A\times B$ such that if $(a,b)\in f$ and $(a,b_1)\in f$ then $b=b_1$ and there exists a $(a,b)\in ...
5
votes
2answers
466 views

What is an ordered pair actually?

What does $(a,b)$ mean actually? I saw this in the 'formal defintion' of functions, and it tripped me up. We haven't even defined what an ordered pair is, before using it. Is it just a notation of ...
1
vote
3answers
108 views

Why isn't an injection an iif?

Suppose that $f:X \rightarrow Y$ is a function. Then an injection can be defined as: $\forall x_1,x_2 \in X, f(x_1) = f(x_2) \Rightarrow x_1=x_2$ Why isn't it defined instead as follows: $\forall ...
0
votes
2answers
37 views

Calculate random integer inside a range of real numbers

$$F : \Bbb R \times \Bbb R \rightarrow \Bbb N $$ $$F(\text{minReal},\ \text{maxReal}) = \text{randomInt} \in \left[\text{minReal},\ \text{maxReal}\right] $$ Let $r \in [0, 1)$ be a random value. How ...
1
vote
3answers
74 views

Why do we define functions to be set theoretic objects?

Why do we define functions to be set theoretic objects? Functions are so intuitive, why do we define it in complicated set theory language?
1
vote
0answers
41 views

What is real periodic function

I would like to know what is real periodic function. I understand what is periodic function, but I do not understand what is "real" periodic function.
1
vote
3answers
89 views

Alternative to epsilon-delta definition

Back in Multivariable Calculus, I remember my professor, when explaining why he decided to skip a rigorous definition of limit (the reason was that those of us that would continue in math would go ...
1
vote
0answers
27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
0
votes
0answers
36 views

Motivation for defining a functions codomain

Why not always refer to a functions image, what is the point in specifying some super set of that image and then naming that super set the functions "codomain"? What does explicitly defining a set ...
6
votes
3answers
76 views

Codomain of a function

At high school we were told that a function has a domain and a range, the function maps from the domain to the range. Such that the domain contains all and only the possible inputs and the range ...
1
vote
1answer
56 views

Precise definition of oscillation behavior of functions like $\sin(\frac1{x})$

I tried today precisely defining the oscillation behavior present in functions like $\sin(\frac1{x})$ i.e: To do this, I started with the domain of function as in limits, and let $f(x)-L$ have ...
0
votes
1answer
18 views

Beginning Proof on functions and Sum of functions

I've been having a hard time with this proof because I do not know where to go from where I am. The proof we were assigned to in class is as follows: Let $f : \mathbb Z \rightarrow \mathbb Z$ be a ...
0
votes
1answer
100 views

What are some practical uses of functions? [closed]

Functions are basically formal equations that relate a set of inputs to output. What are some practical uses for functions and inverse functions?
1
vote
3answers
99 views

Notation for Partial Functions

Suppose we have sets $f$, $A$ and $B$ such that $f\subset A\times B$ and $\forall x\in A\space \forall y,z\in B: [(x,y)\in f \land (x,z)\in f \implies y=z]$ i.e. $f$ is a partial function mapping ...
2
votes
2answers
72 views

Odd and even functions.

I have a book which says: If a function $f$ satisfies $f(-x)=f(x)$ for all $x$ in its domain, then $f$ is called an even function. However, if $f(-x)=-f(x)$ for every $x$ in the domain of $f$, ...
3
votes
1answer
29 views

Linear function definition

I'm trying to figure out what is linearity and what is a linear function. But the wikipedia page confused me. Firstly it defines as polynomial : $f(x) = ax +c$ Than it defines as linear map: ...
1
vote
1answer
18 views

'Union' of maps

Let $f : A \to Y$, $g : B \to Y$. Suppose that $f(x) = g(x)$ whenever $x \in A \cap B$. Define $$ h : A \cup B \to Y, \\ h(x) = \begin{cases} f(x) & \text{ if $x \in A$} \\ g(x) & \text{ if ...
0
votes
1answer
30 views

Grammar question - defining something additionaly

Let's say that I have a function that is defined on some intervals, and on some it's not. I'd like to say that the interval which was not defined, was defined additionaly (because others were already ...
-3
votes
1answer
85 views

Alternative definitions of functions requiring non-empty domains?

It is easy enough to prove in set theory, but it seems counter-intuitive to me that an empty set could be the domain of a function. Is there any literature requiring that functions have non-empty ...
5
votes
3answers
182 views

What is the name of this function $f(x) = \frac{1}{1+x^n}$?

$f(x\in\mathbb{R}) = \frac{1}{1+x^n}$
2
votes
2answers
127 views

Proving that $f(x) = \cos(x)\implies f'(x) = -\sin(x)$ using the definition of a derivative

I'm having trouble grasping the concept which proves that the derivative of $f(x) = \cos(x)$ is $f'(x) = -\sin(x)$. It needs to be proven using the definition of a derivative--and I can't quite piece ...
1
vote
1answer
79 views

Definition of functions on metric spaces.

In the post Definition of functions, it is stated in the accepted answer that one way to define a function is to define it as the triple $(f, X, Y)$ where $f \subset X \times Y$. My question is what ...
1
vote
2answers
91 views

Notation and terminology for functions, interpreting $f(y)$

It seems to me there are two different interpretations of a symbol $f(y)$. I will explain what I mean: Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example). Also ...
1
vote
2answers
48 views

Matrix with Functions as Entries

What do we call a matrix with functions as entries? $$\textbf{f(x)}=\begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix} $$
-3
votes
1answer
59 views

What is the definition of the word many-to-one function?

Does anyone know the definition for the word many-to-one function?
4
votes
1answer
244 views

Set-builder notation function definition

I know that a function is a subset $f \subseteq X \times Y$ such that \begin{eqnarray} \forall x \in X, \exists ! y \in Y | (x,y) \in f \end{eqnarray} First, is it possible to express what a ...
0
votes
0answers
145 views

Rudin's definition of continuity in terms of pre-images (inverse images). Is this simple function continuous or not?

I am reading W. Rudins book ``Principles of Mathematical Analysis''. I find it hard to exactly understand the definition of continuity in terms of pre-images. Rudins definition of a continuous ...
3
votes
1answer
92 views

How to define $a^x$?

It's so common that we use the function $f(x)=a^x$. But actually how do we define it? In simple language we can say $a^n$ is the number $a$ multiplied with $a$ $n$ times for any $n$ in $\mathbb{N}$ ...
0
votes
1answer
105 views

Is this particular function not well defined?

I was looking more into what it means for a function to be well defined, and I believe I understand it. Suppose we have a function $f:A \rightarrow B$ where $A = \{1,2,3,4\}$ and $B = \{1,2,3\}$ ...
1
vote
1answer
54 views

Definition of a function's domain and co-domain with subscript in name

I want to define a function that takes a parameter (lets say a real number) and returns a number (lets say a natural number). However, the function makes use of a 'global environment constant ...
1
vote
1answer
61 views

Is it legal to define a piecewise define a function like this?

I'm trying to piecewise define a function $h$ using two other functions $f$ and $g$. I want to use $h$ to draw conclusions on a certain set $T$ that's a union of two other sets $A$ & $B$. $ ...
3
votes
4answers
2k views

Continuity on open interval

A function is said to be continuous on an open interval if and only if it is continuous at every point in this interval. But an open interval $(a,b)$ doesn't contain $a$ and $b$, so we never ...
19
votes
12answers
4k views

How to represent the floor function using mathematical notation?

I'm curious as to how the floor function can be defined using mathematical notation. What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater than ...
1
vote
1answer
2k views

Definition of correspondence

A one-to-one correspondence is an alternative name for a bijection between two sets, but to what does the term 'correspondence' alone refer? As far as I can see, it seems to be another term for ...
7
votes
1answer
463 views

Can a function be increasing *at a point*?

From what I understand we say that a function is increasing on an interval $I$ if $$ x_1 < x_2 \quad\Rightarrow\quad f(x_1) < f(x_2). $$ for all $x_1,x_2\in I$. I understand that some might call ...
2
votes
1answer
80 views

Is there a common name for the property $f(\lambda x+(1-\lambda)y)\leq f(x)+f(y)$?

For $X$ a vector space and $f:X\to\mathbb{R}$, I'm considering the property $$f(\lambda x+(1-\lambda)y)\leq f(x)+f(y) \qquad\forall x,y\in X, \;\forall\lambda\in[0,1].$$ So it's like convexity ...
0
votes
2answers
38 views

Defined amount and value amount of a function

What is the defined amount and value amount for this function: $$f(x)=\sqrt{(x+7)(1-x)}?$$ The defined amount is all the x-values the function can be and the value amount is all the y-values the ...
1
vote
0answers
185 views

Different Definitions on the Differentiability of Functions on a closed set.

I have encountered three different definitions on the differentiablity of functions on a closed set. In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
2
votes
2answers
356 views

Formal Dirichlet-Bourbaki definition of function

What is the formal Dirichlet-Bourbaki definition of a function? I have come across this in this essay: http://www.k-12prep.math.ttu.edu/journal/contentknowledge/meel01/article.pdf on page 1. I know ...
8
votes
3answers
80 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 $f$ ...
1
vote
1answer
348 views

complement of a function $f: \{2n | n\in \mathbb{N}_0 \}: n \rightarrow n+1$

i am reading a textbook here and i saw, there is notion of Complement of a function. or Negation of a function definiton, this is whow i understood but it is definitely wrong how i do it, i know. in ...
2
votes
1answer
894 views

What is the difference between a function and a map? [duplicate]

Possible Duplicate: Is there any difference between mapping and function? I am an aspiring mathematician who just started out. What is the difference between a function and a map? Or are ...
2
votes
1answer
481 views

Functional independence

Definition confusion: I wish to show that $$f(x,y)={-y\over x}$$ and $$g(x,y)=\log |x|$$ are functionally independent on some domain. What does that mean? What do I have to show? And how does one ...
1
vote
2answers
214 views

visualisation of pointwise boundedness

A sequence of continuous functions $(f_n\colon[a,b]\to\mathbf{R})_{n}$ is said to be point-wise bounded if for all $x\in[a,b]$ there is a $R_x>0$ such that $$|f_n(x)|\le R_x\quad\mbox{for all }n.$$ ...
4
votes
3answers
582 views

well-defined functions

I am asked to argue whether or not the following two functions are well-defined (textbook definition: a) define $y$ for all $x$ in domain, and b) any is mapped to exactly one y). Both of the below ...
2
votes
1answer
292 views

Perfect Hash Function just an Injection?

I just read up on the concept of perfect hash functions on a set $S$. I am quoting: "A perfect hash function for a set S is a hash function that maps distinct elements in S to a set of integers, with ...
7
votes
1answer
505 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...