2
votes
1answer
36 views
Does $f(x)=x^{2}\sin\left(\frac{1}{x^2}\right)$ satisfy the relation $f(x)+f(y)−2f\left(\frac{x+y}{2}\right)=O\left(\left|x−y\right|^2\right)$?
Does $f(x)=x^{2}\sin\left(\frac{1}{x^2}\right)$, $x\in(0,1)$ satisfy the relation $f(x)+f(y)−2f\left(\frac{x+y}{2}\right)=O\left(\left|x−y\right|^2\right)$?
1
vote
1answer
37 views
Does $f(x)=x^{2}\sin(\frac{1}{x^{2}})$ satisfy the relation $f(x)+f(y)-2f(\frac{x+y}{2})=O(|x-y|^{2})$?
Does $f(x)=x^{2}\sin(\frac{1}{x^{2}})$ satisfy the relation $f(x)+f(y)-2f(\frac{x+y}{2})=O(|x-y|^{2})$?
I can't check it. Who will hint it? Please.
2
votes
0answers
46 views
Symmetry between differentiation and integration [duplicate]
I want to make clear, that I am interested in the question: Why does integration need a bigger spectrum of functions than differentiation and not why integration is harder!!!
as experience told me, ...
7
votes
3answers
101 views
Does $f, f' \in L^1([0, \infty))$ imply that $\lim_{x \to \infty} xf(x) = 0$?
Does $\int_0^\infty |f(x)| \, dx$ and $\int_0^\infty |f'(x)| \, dx$ being finite imply that $\lim_{x \to \infty} xf(x) = 0$?
(Context: I am working through an analytic number theory textbook. In a ...
1
vote
1answer
58 views
Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k
In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
0
votes
1answer
42 views
Continuous function on a closed set
Let $f: F \to \mathbb R$ be defined in a closed set $F \subset \mathbb R$. Show that $f$ is continuous if and only if for all $c \in \mathbb R$, the sets $E[f \le c]=\{x \in F; f(x) \le c\}$ and $E[f ...
1
vote
0answers
44 views
Antiderivative of an absolute function
$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$
$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to ...
0
votes
0answers
27 views
If a function $f:J\to\mathbb{R}$ satisfies the Zygmund condition, is it $C^1$?
A function $f\colon J\rightarrow \mathbb{R}$ on an open interval $J$ satisfies Zygmund condition if, for all
$x,y\in J$, $$f(x)+f(y)-2f\left(\frac{x+y}{2}\right)=o(|x-y|).$$ It is clear, if $f\in ...
0
votes
1answer
48 views
Show $C\geq \mathrm{max}\left \{ A,B \right \}$.
Let $\sum_{n=0}^{\infty}a_{n}x^{n}$ and $\sum_{n=0}^{\infty}b_{n}x^{n}$ be the power series with the convergent of radius respectively $A>0$ and $B>0$.
Define $c_{n}=\mathrm{min}\left \{ ...
0
votes
0answers
43 views
Uniform convergence of $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$.
Given that the series $\sum_{n=2}^{\infty}\dfrac{\mathrm{ln}(n)^{p}}{n}$ is convergent iff $p<-1$.
Show that $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$ is uniform convergent as $x \in ...
0
votes
1answer
41 views
Find for all value of constant $a>0$; the interval of convergence of the power series $\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$
Find for all value of constant $a>0$; the interval of convergence of the power series
$\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$.
What I have tried is; if we let $b_{n}=\frac{1}{1+a^{n}}x^{n}$ so ...
5
votes
1answer
83 views
Continuous function differentiable on $[0,1]\setminus\mathbb{Q}$, but nondifferentiable on all of $\mathbb{Q}\cap[0,1]$?
I'm trying to work out an example of a continuous function which is differentiable at all irrationals but nondifferentiable at all rationals in $[0,1]$.
Since $\mathbb{Q}$ is countable, list it as ...
1
vote
0answers
28 views
What's the need of $^{S}_{T}$ in $f^{S}_{T}:S\rightarrow T$?
I'm reading Lang's Undergraduate Analysis:
In the chapter about mappings, he says that we should denote the set of arrival and the set of departure with the following notation:
...
1
vote
2answers
222 views
A less known definition of the definite integral of a continuous function
The definite integral of a continuous function can be defined using the bounded monotone sequence property: see Osgood's Functions of Real Variables, p.110.
(link to full book) (screenshots: page ...
4
votes
1answer
66 views
$\iint f(x,y)\,dxdy$ and $\iint f(x,y)\,dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\!\int_0^1\!f(x,y)\,dxdy$ and $\int_0^1\!\int_0^1\!f(x,y)\,dydx$ both exist, but $f(x,y)$ is not ...
5
votes
2answers
48 views
“Nearly” Harmonic Series
It's well known that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0.
$$
What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$?
...
1
vote
2answers
28 views
Find $y$-Lipschitz constant
$$f(x,y)=x^3e^{-xy^2}, 0\leq x\leq a, y\in \mathbb R, a>0$$
I need to find $K>0$ such that $$|f(x,y_1)-f(x, y_2)|\leq K|y_1-y_2|$$ for all $0\leq x\leq a$ and $y_1,y_2\in \mathbb R$
I did this ...
0
votes
0answers
23 views
How to show that the partial derivatives exist
In general , how to show that the partial derivatives of a multivariable function exists without comupting it .
0
votes
1answer
33 views
A particular weak subadditivity
Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, consider the following property.
For all $(x^1, ..., x^n) \in \left(\mathbb{R}^n \right)^n$ such that $f(x^i) \geq 0$ $\forall i \in [1,n]$, ...
3
votes
1answer
54 views
Oscillation and Hölder continuity
I am studying a proof of a theorem. And I have the following situation in the proof:
Consider $\Omega$ is a bounded open set of $\mathbb R^n$ and $u: \Omega \to \mathbb R$ is a function satisfying:
...
1
vote
1answer
33 views
Parseval's identity
How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
1
vote
0answers
47 views
Taylor Expansion of Power Series
Suppose that $\space f:[0,1]\rightarrow \mathbb{R}$ is real analytic and that its power series expansion is:
$\\ f(x)=\sum\limits_{n=0}^\infty a_nx^n$
Prove that there exists an $x_0\epsilon (0,x)$ ...
1
vote
0answers
44 views
Approximating the modified Bessel’s function with a sum of exponentials
I am looking for an approximation for modified Bessel’s function $I_\alpha(f(t))$ (specially $I_0(f(t))$ or at least $I_0(t)$) with a sum of exponential functions. I mean I want to approximate the ...
1
vote
0answers
33 views
Dominated convergence application?
I'm reviewing for an exam and trying to solve Rudin 11.16, which I think (not sure though) is an application of the dominated convergence theorem:
Suppose $\{n_k\}$ is an increasing sequence of ...
1
vote
1answer
66 views
Why is logarithm in BMO
I read that $\log|x|$ is supposedly a typical example of a BMO function. How do I see that it is in BMO actually?
0
votes
0answers
62 views
Metric on the space of Lipschitz continuous functions
Let $X=C^{[0,1]}([0,1])$, the set of Lipschitz continuous functions with domain $[0,1]$.
a. Prove that
$$\rho(f,g) := \sup|f-g|+\operatorname{Lip}(f-g)$$
is a metric on $X$.
Recall that
...
2
votes
1answer
58 views
About measure theoretic interior and boundary
Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery.
I just want to clarify whether these definitions of measure theoretic interior
and boundary are correct. Given ...
0
votes
2answers
67 views
A few problems on sup and nested intervals
I've been doing these 3 problems for a `proof´ oriented class, one i have found a solution (in fact has been asked here before but the threads are all closed), and checked a correct solution in the ...
4
votes
3answers
85 views
Domain whose boundry has non zero volume.
Can There be a domain in $\mathbb{R^n}$, for any $n$ such that some domain has non zero boundry volume? I.E. volume of boundry is non zero?
Motivation:
In some theorems, it is specified that volume ...
2
votes
1answer
62 views
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$)? And is one a subset of the other?
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$) ? And is one a subset of the other?
$\mu$ is the Lebesgue measure.
6
votes
0answers
183 views
A curious theorem by Peano
Let $f$ be defined on $[a,b]$ and there differentiable.
Show that for every $ \epsilon>0 $ there exists a partition $\, a=a_0<a_1<...<a_n=b \,$ of $ \,[a,b] \,$ so that $$\left|\frac ...
1
vote
1answer
97 views
Determining the Lipschitz constant
Determine the corresponding Lipschitz constant of $f(t,y(t))=e^{(t-y)/2}$, where $D=\{(t,y) : 0\leq t \leq 1,-\infty<y<+\infty\}$.
5
votes
1answer
105 views
Algebraic transformations to continuously extend functions
Lately I was browsing through my analysis lecture notes (since right know I'm somewhat rusty in analysis) and the proof that $x \mapsto \frac{1}{x}$ is differentiable at every $x'\neq 0$ captured my ...
0
votes
0answers
67 views
Show $f$ is in $L^1$ (d$\mu$) space and $_X\int f $ d$\mu=\lim_{n\to \infty}\int_X f_n d\mu$
Suppose $f$ is in $L^1$($\mu$). Prove that for each $\epsilon > 0$ , there exists a $\delta > 0$ so that the $\int |f|\mathrm d\mu$ < $\epsilon$ over the set $E$ whenever $\mu(E) < ...
1
vote
3answers
108 views
For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function?
Alternatively, how does the definition of a limit guarantee that if $f(x)$ is not a constant, then a small $\epsilon$ will give me a small $\delta$?
1
vote
1answer
50 views
What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$?
Let $Y = C^1[a,b]$. What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$.
a) $D = \{ y \in Y: y'(a) = 0,\, y(b) = 1\} $
b) $D = \{y \in Y : ...
2
votes
1answer
36 views
Differentiability of function $f(x,y) = |x|^a + |x-y|$.
I am trying to figure out the points at which the function $f(x,y) = |x|^a + |x-y|$ is differentiable. Could you please help me out. I have considered the cases x>0, y>0 etc, but am having difficulty ...
1
vote
1answer
88 views
Prove that to each $\epsilon >0$, there exists a $\delta >0$ so that the Lebesgue integral…
Suppose $f$ is in $L^1$ space of $\mu$, where $\mu$ is the Lebesgue measure. Prove that to each $\epsilon >0$, there exists a $\delta >0$ so that the Lebesgue integral of the absolute value of ...
1
vote
1answer
41 views
properties of riemann integral real analysis
I need help with this problem please
Suppose that $f$ is Riemann integrable on $[a, b]$ and define the function $F(x) = \int
f(t)dt$ from $a$ to $x$.
Show that $F$ satisfies a Lipschitz condition on ...
-3
votes
0answers
65 views
L'Hopitals Rule [closed]
I need help with this differentiation problem.
Evaluate each of the following limits.
(a) $\displaystyle \lim_{x\to 0} \frac{e^x − cos(x)}{x}$
(b) $\displaystyle \lim_{t\to 0} \frac{sin(t) − ...
-1
votes
2answers
23 views
continuous functions: extremal properties
Please help me with this question.
Let $f : R → R$ be a continuous function that is periodic in the sense that for some number $p$, $f(x + p) = f(x)$
for all $x ∈ R$. Show that $f$ has an absolute ...
-1
votes
1answer
29 views
Ordered sums: series, in real analysis
I need help with this analysis question.
Find all values of $x$ for which the the following series converges and determine the sum:
$\displaystyle x + \frac {x}{1 + x} + \frac{x}{(1 + x)^2} + ...
-1
votes
2answers
32 views
monotone convergence in real analysis
I need help with this analysis problem.
Define a sequence ${t_n}$ recursively by setting $t_1 = 1$ and $t_n = \sqrt{(t_{n-1} + 1)}$. Does this sequence converge? To what?
2
votes
2answers
39 views
Holder condition for $x^\beta$
Let $f(x)=x^\beta$ (for some fixed $0<\beta<1$) be defined on $(0,1)$. It's not hard to see that $f$ is $\beta$-Holder.
How can I prove that $x^\beta$ is not $\alpha$-Holder for ...
3
votes
0answers
91 views
Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$
Suppose $f$ is continuous and periodic on the reals with period 1. Prove that if $x\in[0,1]$ is an irrational number, then
$$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$$
...
1
vote
1answer
154 views
Show that the linear operator $(Tf)(x)=\frac{1}{\pi} \int_0^{\infty} \frac{f(y)}{(x+y)} dy$ satisfies $\|T\|\leq 1$.
Show that the linear operator $T$ given by $(Tf)(x) = \frac{1}{\pi} \int_0^{\infty} \frac{f(y)}{(x+y)} dy$ is bounded on $L^2(0, \infty)$ with norm $||T|| \leq 1$.
The professor also wrote, ...
1
vote
1answer
130 views
+100
A robust convex optimization problem
Consider a function $ f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R} $ such that $\forall x \in \mathbb{R}^n$ the map $f(x,\cdot)$ is convex, and $\forall y\in \mathbb{R}^m$ the map ...
1
vote
2answers
47 views
$|f(x)-f(y)|\ge c|x-y|$, show that $Jf(x)\ne 0 \forall x\in \mathbb{R}^n$ and $f(\mathbb{R}^n)=\mathbb{R}^n$
$f$ is of class $C^{(1)}$ and there exist a number $c>0$ s.t. $|f(x)-f(y)|\ge c|x-y|$ $\forall x,y\in \mathbb{R}^n$. Show that $Jf(x)\ne 0 \forall x\in \mathbb{R}^n$ and ...
0
votes
1answer
30 views
Integral over a Hypercube
Let $C_R$ be a hypercube in $\mathbb{R}^n$ of side length $2R$ and $B_R$ a ball in $\mathbb{R}^n$ of radius $R$.
I know I can say that
$$\int_{B_R}f(x)\,dx \sim \int_{C_R} f(x)\,dx $$
Could anyone ...
2
votes
3answers
32 views
Uniform Convergence and differentiable functions
I have been working on this textbook question and am not sure what to do. Is there a sequence of differentiable functions on some interval, say [0,2], converging to 0 uniformly, but where $f'_n(1)$ ...
