All Questions
1,375
questions
0
votes
2
answers
466
views
Approximation of Sobolev Function in $W^{k,p}(\Omega)$ by Continuous Functions of limited Regularity
Is $C^{k+1}(\Omega)\cap W^{k,p}(\Omega) $ dense in $W^{k,p} (\Omega)$? Assuming $\Omega$ has a $C^1$ boundary (or Lipschitz continuous boundary, if allowed?). I know the standard results on smooth ...
0
votes
1
answer
36
views
Upper bound of $2\vert\cos{\frac{x+y}2}\sin{\frac{x-y}2}\vert$
We have $2\left|\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)\right|$
As $\left|\cos\left(\frac{x+y}{2}\right)\right|\leq 1$
$\left|\sin\left(\frac{x-y}{2}\right)\right|\leq 1$
Is it ...
- 851
1
vote
1
answer
36
views
Can a non-constant continuous function be constant on these hyperbolas?
Can a non-constant continuous function $f:\mathbb{R}^2\to\mathbb{R}$ be constant on the following hyperbolas?
$$H_a=\{(x,y)\in\mathbb{R}^2:x+1/y=a\},a\in\mathbb{R}$$
$$H_\infty=\mathbb{R}\times\{0\}$$
...
- 32.6k
2
votes
1
answer
1k
views
Ratio of convex functions with dominating derivatives is convex?
Let $f,g:\mathbb [0,\infty)\rightarrow (0,\infty)$ satisfy $f^{(n)}(x)\geq g^{(n)}(x)>0$ for all $n=0,1,2,\ldots$ and $x\in [0,\infty)$. In particular, $f\geq g> 0$ are increasing and convex (...
CommunityBot
- 1
0
votes
1
answer
38
views
Constructing a Continuous Function Below an Increasing Function
Let $f$ be an increasing function defined on $[0,1]$ with $f(0)=0$ and $f(x)>0$ for $x>0$. Does there exists a continuous function $g$ on $[0,1]$ such that $g(x)>0$ on $(0,1]$ and
$$g(x)\leq ...
- 368
20
votes
7
answers
26k
views
$f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$
Find a function $f(x)$ such that:
$$f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$$
with $f(4)=65$.
I have tried to let $f(x)$ be a general polynomial:
$$a_0+a_1x+a_2x^2+\ldots a_nx^n$$
which leaves $f(\frac{...
- 2,792
3
votes
0
answers
87
views
Find the values of $b$ for which $f(x)=x^3+bx^2+3x+\sin(x)$ is bijective
Find the values of $b$ for which $f(x)=x^3+bx^2+3x+\sin(x)$ is bijective.
As we know $f(x)$ is surjective, the only task left to prove it bijective is to prove that $f(x)$ is strictly monotonic (...
- 50.3k
2
votes
0
answers
49
views
Prove that $(f+g)(x)<t$ ($t\in\mathbb{R}$) holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.
I got stuck on this question:
Prove that $(f+g)(x)<t$, $t\in\mathbb{R}$, holds if and only if there is a rational number $r$ such that $f(x)<r$ and $g(x)<t-r$.
I think one direction is ...
- 2,493
3
votes
2
answers
74
views
Prove that $g(x) = \sum_{n=0}^{+\infty}\frac{1}{2^n+x^2}$ ($x\in\mathbb{R}$) is differentiable and check whether $g'(x)$ is continuous.
The function $g(x)$ is a function series, so it is differentiable when $g'(x)$ converges uniformly. So I should just check uniform convergence of $g'(x)$ by using the Weierstrass M-test:
$$g'(x) = \...
- 33.9k
0
votes
0
answers
60
views
How to prove that $f : [0,1] \to [0,1] \times [0,1]$ is continuous?
I'm trying to show that the function
$$ f : [0,1] \to [0,1] \times [0,1] $$
$$ t=0.t_1 t_2 t_3 \dots \mapsto (0.t_1 t_3 t_5 \dots, 0.t_2 t_4 t_6 \dots ) $$
is continuous. My idea was to show that the ...
- 61
-1
votes
2
answers
117
views
Need help with creation of an example [closed]
I've been struggling for days now and I cannot come up with an example of a function $f(x)$ that satisfies the following:
$|f(x)−f(y)| < |x−y|$ for any $x, y ∈ R$
AND
equation $f(x) = x$ does ...
- 9
-2
votes
1
answer
41
views
About the Exponential function
Consider function $y=a^x$, $a>1$, and we need to show that
$$\frac{2(a-1)}{(a+1)} < \ln(a) < -1 + \sqrt{2a-1}$$
My idea is to use $y=a^x = \exp{(x\ln a)}$, then find the derivative of $y$ at $...
- 1,131
2
votes
3
answers
343
views
Question about definition of Sequences in Analysis I by Tao.
Here's the definition of a sequence as laid out in the text:
Let $m$ be an integer. A sequence $(a_n)_{n=m}^\infty$ of rational
numbers is any function from the set $\{n \in \mathbf{Z} : n \geq m\}$ ...
- 11.2k
2
votes
1
answer
48
views
Find all $f:\Bbb R\to\Bbb R$ st for any $x,y\in\mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ equals the multiset $\{xf(f(y))+1,yf(f(x))-1\}$.
Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$.
Note: The ...
- 2,665
2
votes
1
answer
122
views
Topological version of uniform convergence of functions
We have a sequence of continuous functions $\{f_n\}$ on a Banach space $X$ and $f_n(x)\to f(x)$ for each $x\in X$ as $n\to\infty$. Given an open ball $B\subset X$ and $\epsilon>0$, we want to show ...
- 529