All Questions
Tagged with real-numbers functions
183
questions
8
votes
4
answers
2k
views
What is the domain of a division of functions?
This question is about real functions of real variables.
I think that, in general, if the domain of some function $f(x)$ is A, and the domain of another function $g(x)$ is B, then the domain of $(f/g)...
- 113
5
votes
2
answers
168
views
Existence of Function Taking Every Value Uncountably Many Times
I am stuck on the following problem coming from a plane geometry theorem I'm working on. This is a generalization of that problem which might actually prove easier to handle. All the techniques that ...
- 233
2
votes
2
answers
110
views
Is this theorem true?
If $f(x)+f(y)=f(x+y)$, then:
$f(x)=a x$
where $a$ is a constant.
Is the above statement true? Is there a way of proving it?
The application of this theorem is in the last part of page 52 (second ...
- 1,141
0
votes
1
answer
32
views
Can I define a function randomly
$D \subset \mathbb R$ and we have two functions $U: D \to \mathbb R$ and $L: D \to \mathbb R$, with the given property that $\forall x \in D: U(x) > L(x)$.
Because $U(x) \neq L(x)$ there are ...
- 12.8k
2
votes
1
answer
61
views
Define powerset P(f) (P(real numbers)
I'd like some help for clarification, as I have no professional help to ask (and also wouldn't want to pay for it yet).
This is part of a German book on mathematical analysis, I don't want the ...
- 85
0
votes
0
answers
57
views
Finding functions that intersect at the minimum number of points
Let $f$ and $g$ be (non-constant) functions from $\mathbb{R}^d$ to $\mathbb{R}$.
For a point $x \in \mathbb{R}^d$, let us define the set $S_{f,x} = \{y \in \mathbb{R}^d : f(y) = f(x) \land y \neq x \}$...
- 33
0
votes
1
answer
38
views
Class $C$ functions
How do you prove the following:
In general, a $C^k$ function is contained in $C^{k-1}$ for any $k$.
Why is this true? Thanks for helping.
- 5,164
0
votes
1
answer
386
views
Linear Mean Function
I've been looking at functions $f:\mathbb{R}^n \to \mathbb{R}$ which necessarily satisfy the following 3 properties. Given $ a_1, a_2, \dots a_n \in \mathbb{R}^+ $
$\begin{array} { l l } 1. & f(...
user536517
0
votes
1
answer
687
views
Are the following formula 1-1 or Onto Functions?
1) Is the function Cube Root of $\sqrt[3]{{-6x-4}}$ One to One Function if domain is all real number?
IMO, I am assuming this is an 1-1 function because well, 1) This will produce a graph of square ...
- 91
3
votes
1
answer
4k
views
Bijection from the irrationals to the reals
Since the irrationals and the reals have the same cardinality, there must be a bijection between them. Somewhere on this forum I found something like this:
Map all of the numbers of the form $q + ...
- 3,709
0
votes
0
answers
122
views
Metric and imperial units, are the calculations correct?
I've been re-learning metric and imperial conversions and know the basic rules of:
1 ft = 0.3048 m
1 m = 3.2808 ft
-
Metric = feet
Imperial = Metres
I've rounded them down to 0.304 and 3.28 ...
- 129
1
vote
1
answer
81
views
slowest integrable sequence of function
Let $I$ (integrable) be the set of continuous functions $f:\mathbb R_+\to\mathbb R_+$ that are integrable and nonincreasing. Let $D$ (divergent) be the set of continuous functions $g:\mathbb R_+\to\...
- 485
14
votes
2
answers
436
views
Solve $f(x+f(2y))=f(x)+f(y)+y$
Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for each $x$ and $y$ in $\mathbb{R}^+$, $$f(x+f(2y))=f(x)+f(y)+y$$
Note:
$f(x)=x+b$ is a solution for all $b\in\mathbb{R}^+$ but I can not prove ...
2
votes
0
answers
661
views
Monotonically increasing function and Lipschitz continuous functions
Let $f(t,x):[0,T]\times \mathbb{R}\rightarrow \mathbb{R}$.
If $$|f(t,x)-f(t,y)|\leq C|x-y|, C>0$$ (Lipschitz continuous functions).
I need to found a monotonically increasing function $g(t;x)$ ...
- 73
0
votes
0
answers
103
views
Domain of function $y$
In my physics book I saw the following math snippet:
Let
$$y(t)=\sin(t)\int_{-\epsilon}^{\epsilon}x(\tau)\cos(t-\tau)d\tau$$
be the output signal for input signal $x(t)$.
So, as a ...
user503399
0
votes
1
answer
496
views
a function defined on the complex numbers
A function $f$ is defined on the complex numbers by $$f(z) = (a + bi)z,$$where $a$ and $b$ are positive real numbers. This function has the property that the image of each point in the complex plane ...
- 4,581
0
votes
3
answers
192
views
Relationship $f(x)$ and $\max(f(x))$
If $f(x),g(x)$ are functions from $\Bbb R$ to $\Bbb R$ and we define $$X=\max\left[0,\max(-f(x))\right]$$ and $$Y=\max\left[0,\max(-g(x))\right]$$
I need to know if the following inequality is true: ...
- 73
6
votes
3
answers
159
views
Does there exist an injection $f:\mathbb{R}\to P(\mathbb{N})$ that satisfies these conditions?
Does there exist an injection $f:\mathbb{R}\to P(\mathbb{N})$ from the reals into the power set of the naturals such that
for any $x\in\mathbb{R}$ the set $f(x)$ is infinite, and
for any distinct $x,...
- 63
0
votes
3
answers
2k
views
The domain of every function is a subset of R.
just wondering if this statement is true or false? And can anyone give an example // Counter Example if the statement is true or false respectively
- 109
4
votes
3
answers
4k
views
Is the Cantor Pairing function guaranteed to generate a unique real number for all real numbers?
I recently learned that for natural numbers, the Cantor Pairing function allows one to output a unique natural number from any combination of two natural numbers. According to wikipedia, it is a ...
- 43
0
votes
2
answers
59
views
Is there an odd function $g$ over the reals such that $g(0)\ne0$ [closed]
Is there an odd function $g$ on domain $\mathbb{R}$, where $g(0)$ isn't equal to $0$ ?
- 1
4
votes
1
answer
177
views
Abstract concept tying real numbers to elementary functions?
Real numbers can be broken into two categories: rational vs. irrational. Irrational numbers can be approximated, but never fully represented by rational numbers.
Analytic functions have Taylor ...
- 247
1
vote
1
answer
222
views
Show function does not have limit using sequence.
Consider the function g: $R$ $ \rightarrow $ $R$ defined by
$$
g(x) = \left\{\begin{aligned}
&4-.5x &&: x\,rational\\
&.5x &&: x\,irrational
\end{aligned}
\right.$$
Pick a ...
- 31
2
votes
1
answer
74
views
Exponential equation
"Let $a, b\in(1, +\infty)$ fixed. Solve the equation:$a^{a^t}=b^{\frac{1}{t^2}\cdot b^\frac{1}{t}}$. "
This problem is from G.M. 3/2017. I can't solve it. For $t\geqslant0$ i showed that there is a ...
- 1,717
0
votes
3
answers
754
views
Proof for sum of two reals
Let $$f(x), \ f(y),\ f(x+y),$$ for $\ x,y \in \mathbb{R}.$
Consider that: $$I) \ \ f(x)=0, \forall x \in \mathbb{R}$$ my professor said that if $I)$ was true, then: $$f(x)=0 \implies f(y)= f(x+y)=0 \ \...
- 127
0
votes
0
answers
42
views
Proving that a certain set of sequences is uncountable
Let $B:=\{(b_1, b_2, b_3, \ldots) : b_i =\pm i!$ for every $i \in \mathbb{N}\}$.
I WTS that $B$ is uncountable. I know there are several ways to do this. At this point I think that constructing a ...
- 4,174
3
votes
1
answer
177
views
Suppose $A$ is a countable subset of $\mathbb{R}$. Show that $\mathbb{R} \sim \mathbb{R} \setminus A $.
Here is the question I am trying to answer: Let $\mathbb{R}$ denote the reals and suppose A is a countable subset of $\mathbb{R}$. Show that $\mathbb{R} \sim \mathbb{R} \setminus A $.
My attempt:
...
- 159
0
votes
1
answer
26
views
Re-definition of scalar product $x \cdot y$ as $\log( 1 + a \, \, x \cdot y)$
Is it possible to set $c_1= c_1(a)$ and $c_2=c_2(a)$ such that for any $a>0$, for any two angles $\theta_i, \theta_j \in [0, 2\pi]$, we have that:
$$
\log( 1 + a \cos(\theta_i - \theta_j) ) = c_1 \...
- 200
0
votes
0
answers
60
views
Find the closest/next $x\in\mathbb{N}$ that satisfies the equation $Ⲭ_{\mathbb{N}}(f(x)) = 1$.
Find the closest/next $x\in\mathbb{N}$ that satisfies the equation $Ⲭ_ℕ(f(x)) = 1$, where $Ⲭ_\mathbb{N}(x): \mathbb{R} \rightarrow \mathbb{N}$ is defined by
$$
\begin{cases}
1 &\text{if } f(x) \...
- 1
1
vote
1
answer
90
views
Find the common level set of two functions
A level set of a real-valued function $f$ of the real variables $x,y$ is a set of the form:
$$L_{x_1}(f)=({(x,y):f(x,y)=x_1})$$
that is, a set where the function takes on a given constant value $x_1$
...
- 3,854
3
votes
1
answer
146
views
Find closest whole integer in equation
Given the following equations:
$$a=\frac{py+qx}{2pq}$$
$$b=\frac{py-qx}{2pq}$$
Where p and q are some real constant number. And $(x, y)$ are some arbitrary real number. Any number can be inputted as ...
- 53
1
vote
2
answers
104
views
Is there a function that produces the decimals of π?
Is there any function Ψ defined for all n ∈ ℕ such that Ψ(n) yields the nth decimal of π? Can such a function exist? How about for any irrational number?
- 852
1
vote
1
answer
54
views
Counter example needed for the following in real analysis
Let {$g_n:[0, \infty) \rightarrow \mathbb{R}$} be a sequence of functions that converges point-wise to $g:[0,\infty)\rightarrow\mathbb{R}$. Suppose $\forall k\in\mathbb{Z}$, $g_n$ converges uniformly ...
- 2,157
1
vote
2
answers
102
views
Why the following sequence of function does not converge uniformly at $[0, \infty)$
Why the following sequence of function does not converge uniformly at $[0, \infty)$ but converge uniformly for some $a>0, [a,\infty)$
$$f_n(x) := n^2x^2e^{-nx}$$
So I know the limit function $f$ ...
- 2,157
2
votes
1
answer
448
views
construct a sequences of integrable function that tends to the dirichlet function.
so I wanted to ask if (it is even possible) to construct a sequence of integrable function $f_n$ such that $f_n \rightarrow f$ where $f$ is the dirichlet function.
$f := \begin{cases}0\ \ x\in[a,b]\...
- 2,157
0
votes
1
answer
113
views
Why is $\{(x, y) \in \Bbb{R}^2 | y = x^2\}$ not a function
$B = \{(x, y) \in \Bbb{R}^2 | y = x^2\}$
Why is this not a function?
I understand that to be a function it must pass the Vertical Line Test,
my only thought is that for every number in $\Bbb{R}^2$ ...
0
votes
1
answer
422
views
show that $f$ is not integrable on $[0, 1]$
Define $f:[0,1] \rightarrow \mathbb{R}$ by
$f(x):=
\begin{cases}
e^{x}, \ x \in \mathbb{Q}\\
e^{-x}, \ x\in \mathbb{Q}^{c}\\
\end{cases}$
show that $f$ is not integrable on $[0,1]$.
I just ...
- 2,157
4
votes
3
answers
5k
views
Is $\frac{1}{x}$ a function?
Consider $f(x)=\frac{1}{x}$ defined on set of real numbers.
If every element in domain has image, then above relation is said to be a function.
But for $x=0$, $f(x)=\text{infinity}$.
Does it mean ...
- 91
1
vote
2
answers
999
views
Is function Injective? Is function surjective? Let R := {r∈R|r>0} and f :R⟶ R given by f(x) = {x if x∈R and x^2 if x∉R}
First time on here! I have no idea why this is giving me so much trouble. I'm definitely overthinking it...
Let ℝ⁺:= {r ∈ ℝ⁺|r > 0} and f :ℝ⁺⟶ ℝ⁺ given by f(x) = {x if x ∈ ℝ⁺ and
...
- 21
0
votes
0
answers
60
views
Different Alternate Representations of Functions
Could someone please point out a source with detailed steps / other pointers for different alternate representations of functions?
For example, I know of two such ways
1) Taylor Series Expansion
2)...
- 800
7
votes
4
answers
1k
views
Is any real-valued function in physics somehow continuous?
Consider the following well-known function:
$$
\operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\
1 & \text{for } x =0 \end{cases}
$$
In physics, the sinc function has ...
- 17.2k
3
votes
0
answers
77
views
$f(f(...f(x)...))$ $a$ times, where $a\in\mathbb{R}$
Take $f(x)$ and do a "double-call":
$f^2(x)=f(f(x))$
I use this notation here to explain my problem.
This can be easy calculated for any function. Also $f^{100}(x)$ is not really a problem. This ...
- 1,545
4
votes
3
answers
1k
views
If $a$ and $b$ are roots of $x^4+x^3-1=0$, $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$.
I have to prove that:
If $a$ and $b$ are two roots of $x^4+x^3-1=0$, then $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$.
I tried this :
$a$ and $b$ are root of $x^4+x^3-1=0$ means :
$\begin{cases}
a^4+...
- 217
0
votes
0
answers
33
views
Given 2 real numbers $a < b$ , let $d(x,[a,b]) = min\{|x-y| : a \leq y \leq b \}$ for $-\infty\leq x \leq \infty$
Then the function $f(x) = \frac{d(x,[0,1])}{d(x,[0,1])+d(x,[2,3])}$ satisfies
(A) $0 \leq x < \frac{1}{2} $ for every $x$
(B) $0 < x < 1$ for every $x$
(C) $f(x) = 0$ if $2\leq x \leq 3$ ...
- 341
0
votes
1
answer
534
views
Determine if $f=\{(x,y)\mid 2x+3y=7\}$ is invertible. From $\mathbb R \rightarrow \mathbb R$. If it is invert it.
I am thinking this is no, because I am not even sure if this counts as a function? I am unsure how this can be a function if there exist only a few $(x,y)$s that fulfill the equation.
Or does the $\...
- 329
0
votes
1
answer
79
views
Having trouble understanding how to disprove/prove if a formula is a function.
Is $\frac 1{x^2-2}
$ a function from $\mathbb{R}\to \mathbb{R}$? Is it a function from $\mathbb{Z}\to \mathbb{R}$?
I have been thinking about this but, I can't find any example for which you can have ...
- 329
1
vote
1
answer
34
views
How to prove $2^{\sqrt{f(n)}} \in O\ (2^{f(n)})$ if $f:\Bbb{N}\rightarrow \Bbb{R^+}$?
How to prove $2^{\sqrt{f(n)}} \in O\ (2^{f(n)})$ if $f:\Bbb{N}\rightarrow \Bbb{R^+}$?
So we want to prove $\exists c\in\Bbb{R^+}:\ [\exists B\in\Bbb{N}:[\ \forall
n\in\Bbb{N}:\ n\ge B\rightarrow 2^{\...
- 2,756
1
vote
3
answers
797
views
Intrepreting tuples as functions
I have been mulling over this for a while now. I am told $\mathbb R^n$ can be interpreted as a set of functions.
Take $\mathbb R^2$, for example I can see how we might interpret it as a set containing ...
- 3,306
1
vote
2
answers
316
views
Solving the functional equation $ xf(x) - yf(y) = (x - y)f(x + y) $ [duplicate]
I found the following functional equation:
Find all functions $f : \Bbb R \rightarrow \Bbb R $ such that:
$$ xf(x) - yf(y) = (x - y)f(x + y) \text{ for all }x, y \in \mathbb R $$
Could you please ...
- 870
2
votes
3
answers
1k
views
What qualifies as a polynomial?
I have a very simple question regarding the definition of polynomials (with real coefficients).
What I've seen so far in terms of defintions:
A polynomial $p(x)$ is a function that can be written in ...
- 8,820