All Questions
Tagged with real-numbers functions
183
questions
3
votes
1
answer
81
views
Suppose $\sum_{n\ge 1} |a_n| = A<\infty.$ Under what conditions is $\sum_{n\ge 1} \epsilon_n a_n = [-A,A]$, for $\epsilon_n \in \{-1,1\}$?
Consider the space of sequences:
$$
\mathcal{E} = \{\{\epsilon_n\}_{n= 1}^{\infty}: \epsilon_n = \pm 1\}
$$
This can be considered a "random choice of sign" in the probabilistic context, for ...
- 208k
0
votes
3
answers
94
views
Is it true that there is a bijection $[0, 1) \to \mathbb{R}$?
Is there is a bijection from $[0,1)$ to $\mathbb{R}$?
I thought of an instance, $$\frac{\sqrt{x(1-x)}}{x-1}.$$
- 126k
1
vote
2
answers
441
views
Find all real solutions $x$ for the equation $x^{1/2} − (2−2x)^{1/2} = 1$
This is what the answer says:
Note that the equation can be rewritten as $\sqrt{x} − \sqrt{2 − 2x} = 1$,
and the existence of such real $x$ implies that $x$ is larger than or equal to $0$ and $x$ is ...
- 2,341
0
votes
1
answer
179
views
Paring function - Output becomes exponential for big real inputs
I am using a Cantor pairing function that takes two real number output unique real number.
def cantor_paring(a,b):
return (1/2)*(a+b)*(a+b+1) + b
This work ...
- 73k
8
votes
2
answers
1k
views
Info on the locale of surjections from the Natural Numbers to the Real Numbers
On the nlab page for locales, it states that there is locale for the surjections from the Naturals to the Reals. This locale has no points (i.e. elements), since there are no such surjections, but the ...
- 5,853
3
votes
2
answers
385
views
What does $f:\mathbb R \rightarrow \mathbb R$ mean?
This is simply a basic notation question: what is the meaning of
$$f:\mathbb R \rightarrow \mathbb R$$
I imagine it's some sort of function to do with the set of real numbers, perhaps some sort of ...
- 157k
-2
votes
2
answers
116
views
Is it true that $\sqrt{ab}\le \frac{a-b}{\ln a - \ln b}$ for any $a\neq b>0$? [closed]
Is it true that $\sqrt{ab}\le \frac{a-b}{\ln a - \ln b}$ for any $a\neq b>0$?
If so, any thoughts on how to prove this?
- 8,369
0
votes
1
answer
135
views
true or false- continuous functions
I'm having some hard time deciding if those sentences are true or false:
$1$. If $f$ is continuous on $\mathbb{R}$ then if $\left|f(x)-x\right|<1$ for every $x$ on $\mathbb{R}$ then $f$ is getting ...
- 11.9k
0
votes
1
answer
42
views
Explain a confusing bound for the integral of a decreasing function.
I am reading a solution of an exercise. In the solution, it says the following:
Consider $g(x,t):=\frac{x}{(1+tx^{2})t^{\alpha}}$, where $x\in (0,\infty)$, $t=1,2,3,\cdots$ and $\alpha>\frac{1}{2}$...
- 5,031
-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.
- 3,320
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 ...
- 6,113
-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 ...
- 312k
8
votes
2
answers
362
views
Interesting functional equation: $f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$
Solve for the function f(x):
$$f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$$
I'm not able to solve this.
[For instance, I tried solving for $f(\frac{x}{x+f(x)})$, but this doesn't lead me ...
- 64.1k
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)...
- 376k
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 ...
- 2,549