Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus.

learn more… | top users | synonyms

1
vote
1answer
45 views

Subsequences and limit inferior

Suppose $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous function. Let $x\in \mathbb{R}$ and let $(x_n) \subseteq \mathbb{R}$ be a sequence converging to $x$. Let $(y_n)$ be a subsequence of ...
2
votes
1answer
24 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
1
vote
1answer
90 views

Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
0
votes
1answer
17 views

$f\in L^{p}(\mathbb R)\cap C_{0}(\mathbb R); (1<p<\infty), g\in C^{\infty}_{c}(\mathbb R) \implies f\ast g \in C^{k}(\mathbb R)$?

We put, $C_{0}(\mathbb R)=$ The space of continuous functions on $\mathbb R$ vanishing at $\infty$; $C^{k}(\mathbb R)=$ The space of all functions $\mathbb R$ whose derivative of order $\leq k$ exist ...
-1
votes
0answers
17 views

Cardinality of the following set of functions on $\mathbb R$ [duplicate]

Consider the following set $W$ = The set of constant functins on $\mathbb R$. $X$ = The set of polynomial functins on $\mathbb R$. $Y$ = The set of continous functins on $\mathbb R$. $Z$ = The set ...
1
vote
1answer
33 views

Proof that this set is not compact

Let $X=C[0,1]$ with the $\sup$ norm. Let $Y = \{f\in X\mid \|f\|_\infty \le 1\}$. It is my goal to show that $Y$ is not compact using the sequence defintion of compactness. Note that it is very easy ...
2
votes
2answers
101 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
4
votes
3answers
109 views

Prove $ne^{-n}$ converges to zero

How would I prove that $ne^{-n}$ converges to zero? I've tried $ne^{-n}<{\epsilon}$ and then logging both sides but no further progress could be made. Thanks
1
vote
1answer
24 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge ...
2
votes
1answer
31 views

Does the limit $\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$ always exist?

Let $f$ be a Lebesgue integable function. Does the limit $$\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$$ always exist?
2
votes
1answer
50 views

There exist $c(c\in \left(\frac{1}{2},1\right)$ s.t $2 \int _{0}^{1}f(x)dx=\frac{f(c)}{c} $

I would appreciate if somebody could help me with the following problem: Let $f(0)=0,f(1)=1,f'(x)>0,f''(x)<0$ on $[0,1]$ Q: Under the proposition is true of false? There exist $c(c\in ...
1
vote
0answers
15 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
-4
votes
0answers
31 views

real anaylsis for twice differentiable function [on hold]

Let f:[0,1] tends to [0,1] be any twice differentiable function satisfying f(ax+(1-a)y) less then or equals to a(f(x)+(1-a)f(y)) for all x,y belongs to [0,1] and any a belongs to [0,1].then for all x ...
0
votes
1answer
52 views

Doubt on an ODE problem

Consider the following differential equation $$x'(t) = h(x(t))$$ Consider a function $x(t)$ which satisfies the differential equation for $0 \lt t \leq 1$ and another function $y(t)$ for $0.5 \leq ...
3
votes
1answer
24 views

Prove this function is absolute continuous and Lipschitz of order $\alpha$

Let $1≤p≤\infty$ and $ f \in L^{p}(a,b)$ such as there is a function $g \in L^{p}(a,b)$ that for all $\phi \in C^{1}(a,b)$ (and continuos in $[a,b]$) with $\phi(a)=\phi(b)=0$ we have: $\int_{a}^{b} ...
-1
votes
1answer
50 views

If $f$ is in $R[a,b]$, show that [on hold]

If $f$ is in $R[a,b]$, show that $$\int_a^b f =\lim_{c\to a^+} \int_c^b f$$ the hint is to show $$\left|\int_a^b f - \int_c^b f \right| = \left|\int_a^c f \right| \le \int_a^c |f|$$
3
votes
2answers
26 views

radius of convergence when root test fails

I'm stuck on this problem: Find the radius of convergence of $\sum \limits_{n=1}^\infty \frac{x^n}{((2+(-1)^n)^n} $ An attempt: From the root test, it seems $L$ does not exist: $$L= \lim_{n ...
0
votes
1answer
28 views

Cauchy Sequence of Differentiable Functions Implies Cauchy Sequence of Derivatives?

Can someone provide a proof of the fact that if I have a sequence of differentiable functions that is Cauchy and converges uniformly on some interval $[a,b]$, then the sequence of their derivatives ...
2
votes
1answer
49 views

Proof: $(\sup(A) - \epsilon)^n<y<(\sup(A)+\epsilon)^n$

Prop.: let be $y \in \Bbb{R}_{>0}$, $n \in \Bbb{N}_{>0}$, and $A \subseteq \Bbb{R}$, then: $$A=\{x| x \in \Bbb{R}_{>0}\wedge x^n \leq y \} \Rightarrow (\sup(A) - \epsilon)^n< ...
0
votes
2answers
48 views

Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by $f(x)=x-[x]$. Determine those points at which $f$ has a limit, and justify your conclusions.

$[x]$ denotes the largest integer that is less than or equal to $x$. Ok, so I can see that $f$ has a limit at every point in $\mathbb{R}\setminus\mathbb{Z}$, but I am having a difficult time ...
2
votes
4answers
113 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...
1
vote
1answer
21 views

Continuity of a function defined by an integral, when the variable is in the region of integration

Hi everyone: Suppose that $f$ is locally integrable in $\mathbb{R}^{n}$, $(n\geq2)$; $B(a,r)$ is the ball of center $a$ and radius $r>0$ , and $\lambda$ is the $n$-dimensional Lebesgue measure. It ...
1
vote
0answers
16 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
0
votes
0answers
5 views

existence of symmetrization algorithm

For symmetrization of a Borel set, the construction of an an explicit algorithm (i.e. a sequence of symmetrization steps that will lead to a ball) is an open question. But have we proved the ...
2
votes
2answers
70 views

The trace of an operator

My question is derived from A. Deitmar's book: A First Course in Harmonic Analysis (second edition), p22, Exercise 1.17. Let me rewrite it again: Let $k:\mathbb{R}^2 \rightarrow \mathbb{C}$ be smooth ...
5
votes
1answer
60 views

Visualisation of the smash product

wedge product, join etc. all of them are no problem for my head, but I am really failing to get a visual idea of what the smash product wants to tell me. For example if I take two spheres, I have no ...
6
votes
1answer
49 views

$\int_0^1 (f(x))^n =$ constant, $f\geq 0$, then $f$ is a characteristic function of a measurable set.

$\int_0^1 (f(x))^n =$ constant, $f\geq 0$, then $f$ is a characteristic function of a measurable set. This is the result from question part (a). Now for part (b), will it also hold when the ...
1
vote
3answers
38 views

Dense subsets of $(L^p(\Omega),\|\cdot\|_p)$

The following results hold. Theorem Let $\Omega\subset\mathbb{R}^n$ be an open set. Then $C^0_c(\Omega)$ is dense in $(L^p(\Omega),\|\cdot\|_p)$, if $1\le p<\infty$. Theorem Let ...
4
votes
2answers
52 views

Borel measure supported on $\mathbb{Q}$

Let $\mu$ be a Borel measure supported on $\mathbb{Q} \subset \mathbb{R}$. Must $\mu$ be a sum of Dirac measures?
1
vote
1answer
49 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
2
votes
4answers
134 views

How to finish this proof about compact implies bounded

A set is called compact if every sequence has a convergent subsequence. I am trying to show: If $K$ is compact then it is bounded. (that it is closed was very easy to prove) What I want to do: Let ...
1
vote
0answers
20 views

Proof for Scheffe's Lemma and General Dominated Convergence theorem

While reading this question here about the proof for Scheffe's Lemma, I was confused since someone said the proof in the question was not correct. I thought the argument was fine, and the author only ...
1
vote
0answers
59 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
0
votes
0answers
46 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
1
vote
2answers
37 views

$\dfrac{\partial}{\partial x}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$

I'm trying to prove the following, interesting, relation: $\dfrac{d}{dx}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$ I tried to integrate by parts the RHS, but i don't ...
2
votes
3answers
87 views

Prove that $\lim \limits_{x\to \infty}e^{\frac{\ln(x)}{x}}=1$

How to prove that $\lim \limits_{x\to \infty}e^{\frac{\ln(x)}{x}}=1$? I know that $x$ grows much faster to infinity then $\ln(x)$, therefore the limit equivalent to $e^0 = 1$ but that's not a ...
3
votes
1answer
30 views

Classify the continuous bilinear functional on $L^p \times L^q$.

Let $1<p<\infty$, $1/p+1/q=1$ and let $L(\cdot,\cdot)$ be continuous bilinear functional on $L^p(\mathbb{R}) \times L^q(\mathbb{R})$. The continuity means that if $f_{n} \rightarrow f$ in $L^p$ ...
1
vote
2answers
47 views

Find out the interval where Rolle's Theorem is applicable

Find out the interval for which the Rolle's theorem is valid for the function $f(x)=2x^3+x^2-4x+2$ My attempt : Supposing the interval is $[a,b]$, $f(a)=f(b)$ gives the equation ...
0
votes
1answer
29 views

Extension of $C^k$ functions: lower bounds

Consider $f\in C^k(U)$ for an bounded open set $U\subset\mathbb{R}^d$, such that $0<A\leq|D f|\leq B$ on $U$, and suppose that $U^\prime$ is a smooth bounded open whose closure is within in $U$. ...
0
votes
1answer
28 views

Open and closed equivalence relations

I am looking for canonical examples of open and closed equivalence relations, especially ones that are generated by a continuous functions. Intuitively I think that an open /closed continuous function ...
2
votes
2answers
84 views

Is $\lim_{x\to 0+} \frac{\ln(x)}{\ln(x)} = \frac{-\infty}{-\infty} = 1$

$$\lim_{x\to 0^+} \sin(x)^\frac{1}{\ln(x)} = ... = \exp \left(\lim_{x\to 0^+} \frac{\ln(\frac{\sin x}{x}) + \ln(x)}{\ln(x)}\right)$$ Now, from continuity we can evaluate each term separately. ...
7
votes
2answers
197 views

Proving that $\int_0^1\frac{x \log^2(1-x)}{1+x^2} \ dx = \frac{35}{32}\zeta(3)+\frac{1}{24}\log^3(2) -\frac{5}{96} \pi^2 \log(2)$

Could we possibly prove this result without using the polylogarithm? I know how to do it by polylogarithm means, but I want a different way. Is that possible? $$\int_0^1\frac{x \log^2(1-x)}{1+x^2} ...
0
votes
0answers
13 views

Expression for volume without changing variables

My question is the same as this: Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables However, my solution, while makes perfect sense to me, is slightly ...
1
vote
1answer
17 views

can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
3
votes
0answers
39 views

Measurability of $\int f(x,\bullet) d\nu$ (product space)

I'm self-studying a set of notes on measure theory. For this chapter, the author said that he is not going to prove every theorem, and I have some trouble with this particular one: Theorem 5.15. Let ...
0
votes
0answers
33 views

Question about Implicit function theorem for differentiable maps

I am having a hard time understanding the statement of the implicit function theorem in Edwards' book "Advanced Calculus". Here are the excerpts from the book. My confusion is about the part on ...
0
votes
2answers
25 views

Minima point is a solution point

Consider $$f:\left[0, \dfrac{\pi}2\right] \to \mathbb R$$ defined as $$f(x)=\sup\{x^2,\cos x\}.$$ It is easy to show that $f$ has an absolute minimum point at $x_o \in I$ , but how to show that $\cos ...
-1
votes
3answers
88 views

Differentiation in Banach spaces

Let $E$ be a Banach space, and $F:=L(E,E)$, with $L(E,E)$ the set of continuous linear funtions in $E$. Prove that the function $\exp: F → F$, defined by ...
1
vote
1answer
47 views

A basic confusion in the proof of Picard's existence theorem

In the proof of Picard's existence theorem of solution of ODE I don't understand the following step: Once it proves that the limit of uniformly convergent series is a continuous function then it ...
2
votes
1answer
57 views

The fundamental theorem of calculus in higher dimension

We know that the following result holds in $\mathbb{R}$: if $f$ is continuous on every compact of $\mathbb{R}$, then the function $$x\mapsto\int_{a}^{x}f(t)~\mathrm{d}t$$ ($a$ is a fixed real number) ...