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, and integration through the fundamental theorem of calculus.

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4 views

For differentiable function where $f'(0)=a$ and $f'(1)=b$ we have that for all $c\in(a,b)$ there exists a $y$ such that $f'(y)=c$.

So what I'm trying to prove: Assume a function $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable and $f'(0)=a$ and $f'(1)=b$. Prove that for any $c\in(a,b)$ there exists a $t\in\mathbb{R}$ such that ...
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10 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = ...
2
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1answer
34 views

Showing that $\int fg\le \int g$ implies $f=0$ a.e.

Take $0<p<1$. If $f$ is locally integrable over on $\mathbb{R}$ and $$\Bigg\vert \int fg\Bigg\vert\le \Vert g\Vert_p\tag 1$$ for every $g$ continuous on a set of compact support, then $f=0$ a.e. ...
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2answers
24 views

Need to prove f(x+P)=f(x) for P>0

Let f be continuous on R and suppose that there exists a number P > 0 such that f(x + P) = f(x) for all x in R. Prove that the function is bounded and uniformly continuous on R. Thus far it is ...
0
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0answers
20 views

Continuity of exponentiation (Terence Tao's book-Lemma 6.7.1)

How to use the sequence $(x^{1/k})_{k=1}$ convergent at 1 (Cauchy sequence) for testing whether $eventually\ $ $\epsilon\cdot x^{-M}-close$ to 1 ? How to get the next result: ...
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0answers
6 views

The $L^p$ convergence rate of the tail of the series $\sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} \}2^{-na}$

This a follow-up to the question: Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-ja}$ When $a > 0$, we have $$ \sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} ...
7
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37 views

If $u\in L^1(0,1)$ is nonnegative and $E_n = \int_0^1 x^n u(x) \, dx$, prove $E_{n-k} E_k \leq E_0 E_n$.

$\textbf{Question:}$ Let $ u \in L^1(0,1)$ be a nonnegative function. Define $$E_n := \int_0^1 x^n u(x) dx$$ Prove the following inequality, $\forall n \ge 0$, and $\forall k \in [0,n]$, we have $$ ...
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0answers
11 views

Upper and lower bounds for functional series

Suppose $x\in[0,a]$, $a>1$. Let $g_0(x)\equiv x$, $g_1(x)=(1+x)/2$, and $g_{n+1}(x)=g_1(g_n(x))=g_n(g_1(x))$. Consider $\{\zeta_i(x)\}_{i\ge0}$ where $\zeta_i(x)$ is defined on the interval ...
3
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1answer
68 views

No clear analytic method to prove unique maximum? ($2^{-x}+2^{-1/x}$)

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software. Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the ...
2
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1answer
35 views

Integration over ellipse

$A=\{(x,y)\in \Bbb R^2\mid \frac{x^2}{a^2}+\frac {y^2}{b^2}=1\}$. Find $\int_A (\cos x)y\,dx+(x+\sin x)\,dy$. Can someone please please give a methodological answer? Thanks a lot!
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0answers
16 views

A substitution causes the new integral to evaluate to the negative of the correct value. Where does it go wrong?

I'm evaluating the function \begin{equation} \iint_D y\mathop{}\!\mathrm dA \end{equation} with $D$ the triangle with vertices at (1,1) (3,2) and (2,3). Evaluating the function without substitution ...
1
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1answer
26 views

I need help finishing this proof using the Intermediate Value Theorem?

Let $f$ and $g$ be continuous functions on $[a,b]$ such that $f(a)\geq g(a)$ and $f(b) \leq g(b)$. Prove $f(x_0)=g(x_0)$ for at least one $x_0$ in $[a,b]$. Here's what I have so far: Let $h$ be a ...
3
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2answers
44 views

compute $\lim_{n\rightarrow\infty}\sum_{k=1}^n \sin(\pi \sqrt{k/n}) (1/\sqrt{kn})$

I'm currently studying for an analysis exam and encountered this problem on an old exam: Calculate the limit: $$\displaystyle{\lim_{n \rightarrow \infty} \sum_{k=1}^n \sin \left ( \pi ...
2
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0answers
10 views

Test for uniform continuity

Test for uniform continuity the function $ f(x, y) = (x^2 + y^2)^\alpha \sin{\frac{1}{x^2+y^2}} $ in $ \{ x^2+y^2 > 1\} $ If we consider $ \alpha < 1 $, then $ \lim_{\sqrt{x^2+y^2} \to ...
-1
votes
2answers
47 views

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$?

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$ ? where f is an bijective function and $f(a)=b,f(c)=d,$ I don't understand graph... I can't see on graph this ...
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1answer
24 views

Collection of all partial functions is a set

I'm studying real analysis from prof. Tao's book "Analysis 1" and I'm stuck on the following exercise: "Let $ X $ , $ Y $ be sets. Define a partial function from $ X $ to $ Y $ to be any function $ ...
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2answers
66 views

Graph connected does not imply $f$ is continuous [on hold]

Show an example of a function $\newcommand{\R}{\mathbb{R}} f: \R \times \R\to \R$ such that $f$ is not continuous, but its graph $$ \Gamma_f := \left\{\bigl((x, y), f(x, y)\bigr) \mid \text{$(x, y)$ ...
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0answers
18 views

Is this infinite series (containing exponentials) uniformly convergent?

Let $a_k >0$ be an increasing sequence of real numbers. Let $f_k$ be real numbers (positive or negative or zero). Let $R \in (0,\infty)$ and fix $y \in (0,\infty)$. Let $$v_n(R) = \sum_{k=1}^n ...
3
votes
1answer
41 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
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1answer
15 views

$L^p$ spaces and proper inclusion

Let $1≤p < q$. Prove that $L^p(\mathbb{R}) \subset L^q(\mathbb{R})$ and the inclusion is proper. I am unsure how to begin this or even prove it about $L^p$ spaces and Banach spaces.
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0answers
33 views

Integration of a function of two variables

How can we check the integrability of $f$ defined on $[0,1] \times [0,1]$ as $f(x,y)=$\begin{cases} 0 & x=\frac{1}{2},y \in \mathbb Q \\ 1 & x=\frac{1}{2},y \in \mathbb Q^c \\ ...
5
votes
1answer
38 views

Inclusions of $\ell^p$ and $L^p$ spaces

I remember seeing this some time ago, but I can't find the examples anywhere. Recall that if $p<q$, then $\ell^p\subseteq\ell^q$ and $L^q[0,1]\subseteq L^p[0,1]$. So we can ask ourselves if any of ...
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4answers
78 views

$\lim_{n\rightarrow \infty}(3^n+7^n)^{1/n}$ [on hold]

I was trying to solve this problem about limit and I have some problems. $$\lim_{n\rightarrow \infty}(3^n+7^n)^{1/n}$$ I need some help with this limit please.
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1answer
15 views

A question involving Frechet differentiability

Let $X, Y$ be real normed spaces and $U \subset X$ open subset. In "Nonlinear functional analysis and applications" edited by Louis B. Rall, we have the followint definition (page 115) A map $F : U ...
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0answers
8 views

Covariance and cross spectrum

A bivariate process $(x_t, y_t)$ is called stationary if each component is a univariate stationary process and $cov (x_s , y_{s+j}) =cov (x_t , y_{t+j}), \forall s,t,j$. The autocovariance function ...
2
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0answers
26 views

If $A$ and $f$ are bounded, then $f$ is integrable in the extended sense (?) [Spivak]

I have a problem with one of the theorems in Spivak's Calculus on Manifolds. I will give some background first: An open cover $\mathcal{O}$ of an open set $A \subset \mathbb{R}^n$ is admissible if ...
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1answer
16 views

Show that if $g((x_n)) \rightarrow l$ and $g((y_n)) \rightarrow m$, then $l=m$

Suppose that $g: (a,b] \rightarrow \mathbb{R}$ is uniformly continuous. Suppose that both $(x_n), (y_n)$ are sequence in $(a,b]$ which converge to $a$. Show that if $g(x_n)) \rightarrow l$ and ...
4
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5answers
539 views

If each term in a sum converges, does the infinite sum converge too?

Let $S(x) = \sum_{n=1}^\infty s_n(x)$ where the real valued terms satisfy $s_n(x) \to s_n$ as $x \to \infty$ for each $n$. Suppose that $S=\sum_{n=1}^\infty s_n< \infty$. Does it follow that ...
0
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1answer
26 views

Show that $f$ is uniformly continuous.

Suppose that $F:(a,b] \rightarrow \mathbb{R}$ is continuous and that the limit as $x \rightarrow a$ of $f(x)$ exists. Show that $f$ is uniformly continuous. I am really struggling with this one. HELP ...
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0answers
12 views

Showing a Hilbert Space has a Schauder Basis

Given $e^2$ = {<$x_1$, $x_2$, ...>, $x_i$ exists in the reals numbers such that $\sum_{i=1}^{\infty}$ $x_i$$^2$ < $\infty$. Show that the vectors $e_i$ \, i=1, ... defined by: $e_1$ = <1, ...
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2answers
41 views

Given $n$ points, the difference of $2$ of them is $1/n$ close to an integer

From today's ENS Ulm Math D exam Let $x_1,\ldots,x_n$ be real numbers Prove there exists $i\neq j $ and $h\in \mathbb Z$ such that $|x_i-x_j-h|\leq \frac{1}{n}$ I tried contradiction and ...
2
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1answer
19 views

Prove a family of function is equi-continuos.

Let $f_n(x)=\sin \sqrt{x+4n^2x^2}$ on $[0,\infty)$. (1)Prove that $f_n$ is equi-continuous on $[0,\infty)$. (2)$f_n$ is uniformly bounded. (3)$f_n \to 0$ pointwise on $[0,\infty)$ ...
4
votes
1answer
38 views

If $f\in L^2[0,1]^2$, do we have $\int_0^1|f(x,x)|dx<\infty$?

Let $f\in L^2[0,1]^2$. Does it follow that $$\int_0^1|f(x,x)|dx<\infty$$ By Cauchy-Schwartz inequality $$\int_0^1|f(x,x)|dx\leq \int_0^1|f(x,x)|^2dx = \int_0^1\int_0^1|f(x,x)|^2dxdy$$ So I need ...
3
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1answer
26 views

Are measurable sets closed under projections?

For the following, let us assume that large enough sets to carry the arguments through do exist, i.e. that there are supercompact cardinals or whatever is sufficient. I know that all projective ...
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21 views

Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-na}$

When $a > 0$, it's fairly easy to see that $$ m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-na} \leq C \quad \forall m \in \mathbb{N}, $$ where $C$ is a finite constant independent of $m$. How ...
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3answers
38 views

A question on real valued function

Let $f : \mathbb R \rightarrow \mathbb R$ be a function such that $f(x + 1) = f(x)$ for all $x \in \mathbb R.$ Which of the following statement(s) is/are true? (A) $f$ is bounded. (B) $f$ is bounded ...
2
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0answers
33 views

Finiteness of the set of zeros

Please help me with the following problem: Let $f$ be a continuous function, linearly bounded, i.e. $$x+A<f(x)<x+B$$ We also now, that there exists an $x_0>0$ such that $$f(x)=x+C,\quad ...
0
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2answers
38 views

An example of a continuous function such that M

An example of a continuous function such that $M$ is the maximum of $f$ on $[a,b]$, and that $\{x_i\}$ is a sequence on $[a,b]$ such that $f(x_i)$ converges to $M$ but $\{x_i\}$ is not convergent. I'm ...
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0answers
11 views

The mutual information rate spectrum

Definition: $\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random ...
0
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1answer
24 views

A Quick Question on the Monotone Property of Integrals.

Let $(\Omega,\mathcal{A},\mu)$ be a measure space with $f$, $g\in \mathcal{L}_1(\Omega,\mathcal{A},\mu)$. If for any $A\in\mathcal{A}$ we have $$\int_Afd\mu\geq\int_Agd\mu\space ,$$ show that $f\geq ...
1
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1answer
45 views

Gradient of a vector function

I have a vectorial function $f$, defined on the set of all $n$-dimensional vectors. $f(x) = \log(x^TAx)$, where $\log$ is the natural logarithm, $x^T$ is $x$ transpose and $A$ is a symmetric $n \times ...
4
votes
1answer
41 views

prove that $\frac{1}{x}$ is not uniformly continuous on $(0,1)$

I would like to show that the function $\frac{1}{x}$ is not uniformly continuous on $(0,1)$ using two approaches. First Approach: We have the fact that if a function $f$ is uniformly continuous on ...
0
votes
1answer
57 views

I can use MVT on $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$?

If I can use MVT: $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt=x\cdot f\left(c\right)$ when $x\rightarrow \infty ,\:c\rightarrow \infty $ so we'll have to evaluate $\lim _{x\to \infty }x\cdot ...
1
vote
3answers
82 views

$f(x)$ is Riemann integrable $\Rightarrow$ $\frac{1}{1 + f^2(x)}$ is Riemann integrable

Let f(x) be Riemann integrable on [a,b]. Then there exist $\lim_{x \rightarrow a+0} f(x)$ and $\lim_{x \rightarrow b-0} f(x)$ f(x) has only removable or jump discontinuities. The set of ...
0
votes
0answers
17 views

Predicting equality/inequality of integrals of multivariable functions

Is it possible to predict equality/inequality, of indefinite integrals of multivariable fucntions, over a domain from equality/inequality respectively of those functions over the same domain? Does ...
4
votes
3answers
484 views

A trigonometric proof of an inequality

We have $f(x) = \sin(\cos(\sin(\cos(...\cos x)...))))$, where $5$ $\sin$ and $5$ $\cos$ are side by side. Prove, that $|f(\frac15) - f(\frac{1}{10})| \le \frac{1}{10}$ I simply have no idea how to ...
0
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0answers
12 views

Where $|f| <\infty$ a.e. condition is used in Vitali Convergence Theorem

Vitali convergence theorem_Wiki Here above is a Wiki article about Vitali convergence theorem, which is referred to Rudin, Real and Complex Analysis. And I'm wondering where the fourth condition is ...
2
votes
1answer
32 views

Divergent succession, but with convergent sum average.

An example of a sequence $a_n$ such that: $$a_n\rightarrow\pm\infty$$ but $$b_n=\frac{\sum_{k=1}^{n}a_k}{n}$$ converge.
1
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0answers
37 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
1
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1answer
20 views

Conditions on $f(t)$ so that $\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt$ converges.

Let us consider $$\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt \ \ \ \ (*)$$ for $a,b\in \mathbb R$. If $f\in L^1(-\infty,\infty)$ the integral converges: ...