All Questions
Tagged with real-numbers functions
183
questions
-2
votes
2
answers
50
views
Let $a$ be a real number such that $a > 0$. Show that the function $f : [a, +\infty) \to \mathbb R, f(x) = \tfrac{1}{x}$ is uniformly continuous.
Let $a$ be a real number such that $a > 0$. Show that the function $$f : [a, +\infty) \to \mathbb R, f(x) = \dfrac{1}{x}$$
is uniformly continuous.
- 1
0
votes
1
answer
106
views
Suppose that for every $x$, $y$ such that $x$ is not equal to $y$ we have $|f(x) − f(y)| < |x − y|$
Let $a$ and $b$ two real numbers such that $a < b$ and $f : [a, b] \to [a, b]$.
Suppose that for every $x$, $y$ such that $x$ is not equal to $y$ we have
$|f(x) − f(y)| < |x − y|$. Show that ...
- 27
-1
votes
2
answers
21
views
How to find the value of this composite function: [closed]
Let $ f,g :\mathbb{R} \to \mathbb{R} $ function such that $ f(x + g(y)) = -x+y+1 $ or each pair of real numbers x and y what is the value of $ g(x+f(y)) $ ?
Please help me with some clue. Thanks in ...
- 41
3
votes
2
answers
86
views
If following actions allowed, Find $F(2002,2020,2200)?$
If following actions allowed,Find $F(2002,2020,2200)?$
$$ F(x+t,y+t,z+t)=t+F(x,y,z);$$
$$ F(xt,yt,zt)=tF(x,y,z);$$
$$ F(x,y,z)=F(y,x,z)=F(x,z,y)$$
where x,y,z,t are real numbers.
My attempt:
$F(0,0,0)...
- 59
0
votes
2
answers
89
views
Is there a name for a real-valued function whose input is also real?
I'm trying to write a sentence about a function $f:\mathbb{R}\to \mathbb{R}$, and I want to refer to it as real valued or as a scalar function, or some similar term, but I want that term to also ...
- 1,361
0
votes
2
answers
134
views
Is $\frac{1}{\frac{1}{x}}$ defined at $x=0$?
In the context of projectively extended real line $\widehat{\mathbb{R}}$, if $f(x)=\frac{1}{\frac{1}{x}}$, then
$$f(0)=\frac{1}{\frac{1}{0}}=\frac{1}{\infty}=0.$$
But in the context of $\mathbb{R}$, ...
- 311
2
votes
4
answers
81
views
Find values of $x$, such as $\log_3 \sqrt{x+3}−\log_3(9−x^2) < 0$
The Function is $$f(x) = \log_3\sqrt{(x+3)}−\log_3(9−x^2)$$
and I need to figure out arguments for which $$ f(x) < 0 $$
So I calculated the domain of function which is $ D: (-3;3)$
However I am ...
- 31
9
votes
3
answers
153
views
How different can $f(g(x))$ and $g(f(x))$ be?
Given $f,g: \mathbb{R} \rightarrow \mathbb{R}$, how "different" can $f(g(x))$ and $g(f(x))$ be?
By "how different" I mean:
Given two real-valued functions $a,b$ do there exist two real-valued ...
- 2,471
0
votes
1
answer
73
views
Is there a specific name for $Y=(X-c)^+$, as a function of a random variable?
Let $X$ be a random variable, and for a given $c>0$ let $f_c:\mathbb R\to [0,\infty)$ be a measurable function defined by $x\mapsto \max\{x-c,0\}$. Write $Y:=f_c$.
I have not been able to find a ...
- 37.2k
-1
votes
1
answer
132
views
Is a function $f(x)=\ln({x^2-1})$ even and symmetric
We have a function:
$$
f(x)=\ln(x^2-1)
$$
The function is symmetric because: $D_f=(-\infty,-1) \ \cup\ (1,\infty)$
I understand this as if we would multipy this by $-1$ we would get the same $D_f$
...
- 2,527
1
vote
3
answers
329
views
Does there exist a function which is unbounded in all local neighborhoods?
I have heard of a function which is unbounded for all and any neighborhood of any real X.
I can't seem to wrap my head around the possibility of such a function and my companion can't remember the ...
- 71
0
votes
0
answers
38
views
The Maschler's bargaining set in the cooperative game theory, missing a step in the proof
I have a problem with the concept of the bargaining set which is given below in some detail.
Let $N=\{1,\ldots,n\}$ be a set of players and $v:2^N\to \mathbb{R}$
a superadditive game (meaning $S,T \...
- 3,978
-1
votes
1
answer
52
views
Tool Like Argument Principle For Real-Valued Functions
The Argument Principle gives a way of numerically counting the number of roots-poles ($Z-P$) of a meromorphic function in a contour. I was wondering, can the Argument Principle (or some other tool ...
- 87
5
votes
2
answers
91
views
Finding the number of continuous functions
Question:
Find the number of continuous function(s) $f:[0, 1]\to\mathbb{R}$ satisfying $$\int_0^1f(x)\text{d}x=\frac{1}{3}+\int_0^1f^2(x^2)\text{d}x$$
My approach:
I put $x^2=t$, giving $2x\text{d}x=...
- 127
2
votes
2
answers
208
views
Why is this function continuous on $\mathbb R$?
Let $f:\Bbb{R}\to\mathbb R$ be a function with $f(0) = 1$ and $f(x+y) \le f(x)f(y)$ for all $x, y \in \Bbb{R}$. Prove that if $f$ is continuous at $0$, then $f$ is continuous on $\Bbb{R}$?
THOUGHTS:
...
- 160
1
vote
2
answers
231
views
Set of constant functions are uncountable.
Let $F=$ $\{$ $f: [0,1] \rightarrow \mathbb{R}$ $:$ $f$ is constant$ \} $. I must show that $F$ is uncountable.
Note, that for any $f \in F$, and any $c\in \mathbb{R}$, I will denote the constant ...
user643073
2
votes
0
answers
72
views
Proving integral of a continuous function is continuous
Let $U \subset \mathbb{R}^N$ be an open set, let $f : U \times [a, b] \to \mathbb{R}$ be a continuous function. Consider the function $$g(x):= \int_a^b f(x,y) \,dy$$
with $x \in U$.
i) Prove ...
- 1,913
2
votes
2
answers
461
views
Strictly increasing bounded function of class $C^1$
Let $f:\mathbb{R} \to \mathbb{R}$ be a strictly increasing bounded function of class $C^1$. Prove that there exists a sequence $\{x_n\}_n$ of real numbers such that $x_n\to\infty$ and $\lim_{n \to \...
- 1,913
0
votes
1
answer
77
views
Let $f:\mathbb R\to\mathbb R$ be a function. Prove that the following two statements are equivalent:
(i) $f$ is continuous and satisfies $f(f(x))=x$ for all $x\in\mathbb R$, and there exists $k\in\mathbb R$ such that $f(k)\ne k$.
(ii) There exists a real number $k$ and a function $g:(-\infty,k]\to[k,...
- 4,266
0
votes
0
answers
28
views
Parametric representation of curves
If γ is a $C^1$ curve with parametric representation $φ: [a,b] → R^n$ (i.e. with φ ∈ $C^1$([a, b])) then $L(γ)= \int_a^b ||φ'(t)|| dt$.
I know this hold for $C^1$ but does it also hold for piecewise $...
- 1,913
0
votes
1
answer
171
views
Composition of Continuous functions with a finite number of points
Suppose $f$ is continuous everywhere except for a finite number of points and $g$ is continuous everywhere. Then show $g \circ f$ is continuous everywhere except for a finite number of points. Show ...
- 1,913
0
votes
1
answer
36
views
proving the existence of a real number c such that the function holds
h: R—>R be a function
h(m+y)=h(m)+h(y)
h(0+0)=h(0)+h(0) –>h(0)=0
h(n)=h(1)+h(1)+...+h(1) (n times)
conclude that h(n)=n*h(1)
Since h(x)xH(1)H(1)< H(1)y
hence H(t)=tH(1)
so ∃ c∈R s.t. h(x)=c*x ...
- 553
0
votes
2
answers
33
views
How to read the following mathematical notation?
How to read the following mathematical expression?
$$\text{For any strictly increasing function } f:\Bbb R\to \Bbb R, v(x)=f(u(x))$$
- 51
3
votes
2
answers
90
views
How can i find the domain of $f(x)$= $x^{1/x}$ on the negative numbers?
I have been thinking of which negative values of $x$ yield a real number when plugged on $f(x)$ = $x^{1/x}$,
It is clear to see that it does not work for negative even numbers
$(-2)^{-1/2}$ = $(\...
0
votes
1
answer
192
views
How to find all the continuous functions satisfying an equation? [duplicate]
The problem that I want to solve is:
"find all the continuous functions $f\colon \mathbb R\to \mathbb R$ such that for every $x$, $f(f(f(x))) = x $ , I know that f(x) = x is an answer but how can ...
- 540
1
vote
3
answers
114
views
Decreasing positive function less than $x$?
Is there a decreasing function $f$ defined on $(0,∞)$ such that $0<f(x)<x$?
I thought about it and couldn't come up with a conventional function. I was thinking that for all $x>0$ we can find ...
- 3,134
0
votes
0
answers
53
views
solve for function which is the written in terms of itself and its inverse
Let, $f:\rm I\!R \rightarrow \rm I\!R$ be some function that is equal to a linear combination of itself and its inverse. Is is possible write an explicit formula for $f(x)$?
$$
f(x) = af(x) + bf^{-1}(...
- 85
1
vote
3
answers
1k
views
Prove that $f(x)=\ln\left(\frac{x+1}{x-1}\right)$ is surjective
Prove that $f(x)=\ln\left(\dfrac{x+1}{x-1}\right)$ is surjective.
I found easily that this function is injective, but I need to prove that it's surjective too. The final goal is to prove that this ...
- 3,312
2
votes
1
answer
102
views
Is it possible to find the local maximum of $\sqrt[x]{x}$ without using derivative?
Let $f(x)=\sqrt[x]{x}$, where $x\in\mathbb{R^{+}}$
Using derivative,
$$\frac{d}{dx}(x^{\frac 1x}) = -x^{\frac 1x - 2} (\log(x) - 1)$$
$$f'(x)=0 \longrightarrow x=e$$
$$\text{max}\left\{\sqrt[x]{x}...
user548054
1
vote
0
answers
94
views
Best approximation of a real number with two functions
There's two functions, called $F(x)$ and $G(x)$, where $F(x)>x,1<G(x) < x$ ,$F'(x)>0, G'(x)>0$ on $(2,\infty)$, and $F(x),G(x)\in(2,\infty)$. Given $x, y\in (2,\infty)$, and now I want ...
- 5,054