Tagged Questions
17
votes
11answers
2k 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
92 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 ...
6
votes
1answer
93 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
71 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
30 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
63 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
108 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
75 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
45 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 ...
1
vote
1answer
103 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 ...
1
vote
1answer
131 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
77 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
321 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 ...
1
vote
1answer
209 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
303 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
150 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
240 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
255 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
409 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?
...
3
votes
5answers
1k 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 ...
19
votes
9answers
2k 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 ...
