All Questions
1,375
questions
4
votes
0
answers
115
views
Show that $f(a,b,c)=(a+b+c)^3+(a+b)^2+a$ is injective.
For a function $f : \mathbb{N} \times \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ defined by
$f(a,b,c)=(a+b+c)^3+(a+b)^2+a$
I want to show that $f$ is injective.
How can I show this?
I ...
- 41
0
votes
1
answer
59
views
Real Analysis - discontinuity
Let $f(x)$ = $x^2$ if $x$ is rational and $f(x)$ = $0$ if $x$ is irrational.
Prove that that $x$ is discontinuous at all $x\neq 0$
We look at the case where $x\neq 0$ and $x\in R\setminus Q$
Proof
Let ...
- 150k
17
votes
3
answers
366
views
How do I construct a function $\operatorname{sog}$ such that $\operatorname{sog}\circ\operatorname{sog} = \log$?
Imagine a real-valued semilog function $\DeclareMathOperator{\sog}{sog}\sog$ with the property that
$$\sog(\sog(x)) = \log(x)$$
for all real $x>0$.
My questions:
Does such a function exist?
...
- 9,261
0
votes
0
answers
17
views
When to include the boundary points in the convexity analysis?
I was wondering about this: suppose I have a function $f: D \to \mathbb{R}$; suppose $(a, b) \subset D$ ($D$ can either be bounded or unbounded), and say $f$ is convex in $(a, b)$.
What is the ...
- 3,482
1
vote
3
answers
118
views
Prove that if $f: \mathbb{R} \to \mathbb{R}$ is convex and bounded from above, then $f$ is constant.
Prove that if $f: \mathbb{R} \to \mathbb{R}$ is convex and bounded from above, then $f$ is constant.
Our teacher showed us this, and asked us to solve the rest. But I'm a bit confused by what he did.
$...
- 29.8k
1
vote
1
answer
87
views
Studying the sign of this function without the derivative
I want to study where $f(x) > 0$ and where $f(x) < 0$.
Is there a way we can study the sign of
$$f(x) = \frac{2 x^3-x^2+5 x+4}{-2 x^3-6 x^2+8 x}$$
withouth making use of the derivative? I mean ...
- 2,262
2
votes
2
answers
233
views
How can I prove that a continuous decreasing function on $\Bbb R$ has a fixed point?
I am trying to prove that a continuous decreasing function $f: \Bbb R → \Bbb R$ has a fixed point.
I tried to use the function $g(x) = f(x) - x$, which should be a decreasing one, but I don't know ...
- 52.3k
5
votes
1
answer
217
views
When does the square root of $f:\mathbb{N} \rightarrow \mathbb{N}$ exist? [duplicate]
Let $f:\mathbb{N} \rightarrow \mathbb{N}$. When does there exist a $g: \mathbb{N}\rightarrow \mathbb{N}$ such that $f(n)=g(g(n))$ for all $n$?. I don't think its always possible, for example $f(n)=n+1$...
- 15.9k
2
votes
1
answer
77
views
Show that the equation $(1-x)\cos{x} = \sin{x}$ in $(0,1)$ has at least one solution.
Show that the equation $(1-x)\cos{x} = \sin{x}$ in $(0,1)$ has at least one solution.
We have $f(x) = (1-x)\cos{x} - \sin{x} = 0$
We have
$$f(0) = (1-0)\cos{0} - \sin{0} = 1 - 0 = 1 > 0$$
$$f(1) = ...
- 150k
1
vote
0
answers
55
views
There exists an expansion function from $S_1 \to S_1$ that is continuous and has no fixed points?
If $S_1 = \{ x \in \mathbb{R}^2 : \|x\| = 1 \}$ is it possible to construct a function $f: S_1 \to S_1$ that is continuous and for all $x, y \in S_1$ we have $|f(x)-x| \ge \dfrac{1}{10}$ and $|f(y) - ...
- 59
0
votes
1
answer
62
views
To justify a complex-valued function is continuous
A complex-valued function is defined on the unit disk as $f(z) = \int_{0}^{1} \frac{1}{1-tz} dt$. How can we show that the function is continuous ?
My Approach: As the integrand is analytic in $z$, it ...
- 3,612
8
votes
1
answer
339
views
Prove that $\lim _{n \rightarrow \infty} \sqrt{n} \cdot\sin^{\circ n}(\frac{1}{\sqrt{n}})=\frac{\sqrt{3}}{2}$ [duplicate]
It's known that $\lim _{n \rightarrow \infty} \sqrt{n} \cdot \sin^{\circ n}(x)=\sqrt{3}$ for any $x>0$.
And I found a new conclusion $\lim _{n \rightarrow \infty} \sqrt{n} \cdot \sin^{\circ n}(\...
- 170k
0
votes
0
answers
50
views
How can I prove/disprove that f is a contraction mapping?
How can I prove that this function: $ f: X \rightarrow X \text{ with } X = [1, \infty)$ (The metric is the absolute value)
$f(x) = x+\frac{1}{x}$ is not a contraction mapping?
I think it isn't because ...
- 75
0
votes
1
answer
100
views
Injective and / or surjective: factorial
I would like to know if my reasonment are correct, so thank you in adavance for every critic or missing detail you will point out.
The question is: consider $f: \mathbb{N}\to \mathbb{N}$ definied by $...
- 1,516
1
vote
1
answer
179
views
Inverse image of an open set under a continuous function is the intersection of such set with the domain of $f$
I have been faced with the following problem while studying some notes on $\mathbb{R}^{n}$ analysis:
Let $D \subset \mathbb{R}^{n}$ be a set and $f: D \rightarrow \mathbb{R}$ be a function. Prove ...
- 3,165