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1,376
questions
7
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1
answer
901
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A question about Darboux functions [duplicate]
Possible Duplicate:
A converse of sorts to the intermediate value theorem, with an additional property
Definition of Darboux function: Let $S\subset\mathbb{R}$ be given. We say that $f:S\...
CommunityBot
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3
votes
2
answers
158
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A function $f\in C([0,1])$ with certain properties
Let $f\in C([0,1])$ be differentiable on $(0,1)$. Suppose that $f(0)=f(1)=0$ and that there is an $x_0\in(0,1)$ where $f(x_0)=1$. I have to prove that there is a $c\in(0,1)$ satisfying $|f^\prime(c)|&...
CommunityBot
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2
votes
1
answer
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Families of functions under differential operation
The function $f = \sin x$ is the member of a family of functions closed under the operation $\frac{d}{dx}$. Are there other families of functions that are similarly closed under differentiation? ...
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1
vote
1
answer
193
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Showing that a $2^n$ grows faster than $n^{\log_2 \log_2 n}$
I want to show, that $2^n \in \Omega(n^{\log_2 \log_2 n})$, thus I want to show, that there is a value $n_0$ from which on $2^n$ will always be bigger than $n^{\log_2 \log_2 n}$.
$$2^n \geq n^{\log_2 ...
- 13.2k
5
votes
3
answers
776
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Proving $\Bigl[\frac{8n+13}{25}\Bigr] - \Bigl[\frac{n-12 - \Bigl[\frac{n-17}{25}\Bigr]}{3}\Bigr]$ is independent of $n$
This an Exercise from Apostol's analytic number theory. I have been struggling with this problem for quite some time. Looks elementary though.
The question is to prove that this quantity
$$\Biggl[\...
- 544
10
votes
3
answers
3k
views
For how many functions $f$ is $f(x)^{2}=x^{2}$?
How many functions $f$ are there that satisfy $f(x)^{2}=x^{2}$ for all $x$?
My text (Spivak's Calculus; chapter 7 problem 7) asks this question for continuous $f$, for which the answer is, of course ...
- 620k
2
votes
1
answer
544
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limit of Lebesgue integrable functions
Given two sequences of integrable functions $\{f_{n}\}, \{g_{n}\}$ with limits $f$ and $g$ both also integrable. Does this always hold
$$\lim_{n}(f_{n}-g_{n})=\lim_{n}f_{n}-\lim_{n}g_{n}=f-g$$
I ...
1
vote
0
answers
60
views
find a bound on a particular class of functions
let $a\in(0,1)$, let $f:M\times N\times [0,T)\rightarrow \mathbb{R}$ a $C^{\infty}$ function, with $M,N$ compact manifolds. Assume that we have a closed, convex set $F$ of such functions such that the ...
- 111
4
votes
2
answers
149
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Sequence of functions (convergence)
Let be $f(x)=\frac{2x}{1+x} $ function and $ x_0 > 0 $. With the help of this, form the $x_{n+1}=f(x_n)$ sequence. Is $x_n$ convergent and if yes what is the limit?
Thank you very much in advance!
CommunityBot
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25
votes
6
answers
815
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Is there a function with this property?
Is there a real function over the real numbers with this property $\ \sqrt{|x-y|} \leq |f(x)-f(y)|$ ?
My guess is no but can anyone tell me why?
This came up as a question of one of my collegues and ...
- 42k
2
votes
1
answer
128
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Meaning of $C(I,\mathbb{R})$ and $C^{\infty}(I, \mathbb{R})$ related to continuous functions
What does this mean: ($f$ a function, $I$ an interval and $R$ the real numbers)
$f \in C(I,R)$
Does it mean $f$ is an element of the collection of continuous functions with domain $I$ and range $R$ ?...
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