0
votes
1answer
17 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 ...
1
vote
1answer
46 views

What are some practical uses of functions? [on hold]

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
2answers
68 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
50 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
25 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
14 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
25 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
79 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 ...
4
votes
3answers
149 views

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

$f(x\in\mathbb{R}) = \frac{1}{1+x^n}$
1
vote
2answers
94 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
66 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
79 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
45 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
55 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
212 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
103 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
90 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
85 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
46 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
57 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
740 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 ...
17
votes
12answers
3k 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
1k 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
315 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
79 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
37 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
151 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
285 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
194 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
613 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
347 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
172 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
535 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
256 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
445 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
1
vote
3answers
163 views

What is the proper term for a function where domain and codomain coincide?

What is the proper term for a function where domain and codomain coincide? E.g. in programming languages a function f : Int => Int or f : Double => Double. Thanks.
5
votes
4answers
319 views

Definition of injective function

From wikipedia I obtain the following definition of an injective function : Let $f$ be a function whose domain is a set $A$. The function $f$ is injective if for all $a$ and $b$ in $A$, if $f(a) = ...
1
vote
2answers
681 views

Subexponential growing functions

What is the most common definition of a subexponential growing function ? It seems there are different notions in literature.
8
votes
4answers
589 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? ...
4
votes
5answers
2k views

Difference between kernel and function?

I have been looking around for this question, but all results I found only describe the definition and not the answer I seek. Is "kernel" basically a synonym of "function"? When should be the time we ...
23
votes
10answers
3k views

How do you define functions for non-mathematicians?

I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...