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.

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-2
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1answer
52 views

Show that E has no interior points.

Let f $\in$ C' in the open set $\Omega$ and have no critical points there. Let E be the set where f(p)=0. Show that E has no interior points.
0
votes
1answer
53 views

Is that proof correct? I'm proving that the set of injective Linear transformations in open.

Let $T$ an inyective linear transfomation from $\mathbb{R}^n$ to $\mathbb{R}^m$. Then there exist an $\alpha>0$ such that $$||T(x)||\geq \alpha||x||$$ for all $x$. Let $S$ a linear transformation ...
1
vote
1answer
27 views

Set where function has high values is small

Let $\mu$ be a probability measure on a set $A$, and let $f:A\rightarrow\mathbb{R}$ be a random variable. Given $\epsilon>0$, is it true that we can find $n$ such that ...
1
vote
1answer
95 views

If $||f'(x)||\le M \;,\,\forall x\in (a,b)$ then $||f'(x)||\le M \;,\,\forall x\in [a,b]$

The following proposition is true? Proposition. Let $f:U\longrightarrow \mathbb{R}^n$ a differentiable function where $U\subset\mathbb{R}^m$ $\color{blue}{(m\ge 2)}$ open set and $[a,b]\subset ...
0
votes
2answers
34 views

Two general questions regarding intervals in $\mathbf{R}$

Does an open interval contain infinitely many closed intervals? Is the interval $(-1,1)$ equal to $[-a,a]$ where $a<1$? Why?
2
votes
0answers
17 views

Compute tetrahedral region

Show that the volume of region $A$ is $1/6$. Region $A$ is a tetrahedral region in $\mathbb R^3$. $$A=\{(x,y,z)∈R^3 \mid x\ge 0, y\ge 0, z\ge 0, \text{ and } x+y+z\le 1\}$$
1
vote
1answer
114 views

Absolute value bound of Lebesgue integral

For the Riemann integral, we have the bound $$\left|\int_Af(x)dx\right|\leq\left(\sup_{x\in A}|f(x)|\right)\cdot\left|\int_Adx\right|$$ Do we have a similar bound for the Lebesgue integral, one like ...
2
votes
3answers
100 views

Is series convergent/divergent

I need to find out is series convergent or not $$ \sum_{k=1}^\infty \frac{5k-2}{(3{k}^{2}-2)\sqrt[3]{k+6}} $$ How can I do that? Can you show step-by step solution?
0
votes
2answers
235 views

Why is every continuously differentiable function with a uniform bounded derivative lipschitz continuous

I only know how to prove this for functions on a convex set by using the mean value theorem, but is this also true for this general case when nothing is said about the domain of the function besides ...
0
votes
3answers
43 views

Convergent series: am I doing it right?

I have $$ \sum_{k=1}^\infty \frac{5k-2}{(3{k}^{2}-2)\sqrt[3]{k+6}} $$ I get (i did not write all solution as it is quite hard to me to put this in LaTeX by myself) $$ \lim_{k\rightarrow \infty} ...
1
vote
2answers
28 views

Simple Question about the norm.

Suppose we have a Banach space $V$ with a norm $\|\cdot\|:V\to \mathbb{R}$. Is the following true for all linearly independent vectors $x,y\in V$?: $$\|x+y\|\geq \|y\|,~~~~\|x+y\|\geq \|x\|$$
1
vote
2answers
177 views

Pointwise convergence does not imply $f_n(x_n)$ converges to $f(x)$

I have been given a sequence of real valued continuous functions $(f_1,f_2,...,f_n)$, and a real valued sequences $(x_1,x_2,...,x_n)$, where $x_n$ converges to $x$. Also $f$ is a continuous real ...
2
votes
0answers
34 views

Radius of Convergence and Interval of Convergence: Am I doing it right?

I must find ROC and IOC in $$ \sum_{n=1}^\infty \frac{{(-1)}^n(x^{2n})}{\sqrt[3]{n^2+4n}} $$ I get $$ R= \lim_{n\rightarrow \infty}\left| \frac{a_n}{a_{n+1}} \right| = \cdots = \lim_{n\rightarrow ...
0
votes
1answer
38 views

How to find $\lim_{n\rightarrow\infty}\frac{(n+1)^{k}+(-n)^{l}}{(n-1)^{k}-n^{l}}$ for $k,l\in\mathbb{N}$.

How to find $$\lim_{n\rightarrow\infty}\frac{(n+1)^{k}+(-n)^{l}}{(n-1)^{k}-n^{l}}$$ $k,l\in\mathbb{N}$. I know that it is 1: for $k>l$ for $k < l$ : $$(-1)^{1+l}$$ for k=l odd: -1 for k=l ...
3
votes
1answer
384 views

$f(x)=\sin x^3$ for $x\in \mathbb{R}$ is not uniformly continuous

Question is to prove that : $f(x)=\sin x^3$ for $x\in \mathbb{R}$ is not uniformly continuous. What would my first observation in checking uniform continuity is to check if its derivative is ...
2
votes
2answers
93 views

Proving Continuity for $f(x_1 + x_2) = f(x_1)f(x_2)$

Suppose $f$ is defined on $\mathbb{R}$ and has the following properties. Its limit when $x$ approaches $0$ is $1$ and $f(x_1 + x_2) = f(x_1)f(x_2)$. Then prove that a) $f(x)>0$ for all $x$ b) ...
0
votes
1answer
55 views

Continuity of a constant function

If f assumes only finite many values, then f is continuous at a point $x_0$ if and only if f is constant on some interval $(x_0 - \delta, x_0 + \delta)$ I know how to prove continuity for a given ...
1
vote
2answers
66 views

Question about $f :\mathbb{R}\rightarrow \mathbb{R}$ defined as $f(x)=|x|^{\frac{3}{2}}$ (TIFR GS $2010$)

Question is : I am not sure how to check for differentiability, only thing i know is how to see for differentiability of $f(x)=|x|$ $\lim_ {x\rightarrow 0} \frac{f(x)}{x}=\lim_ {x\rightarrow 0} ...
2
votes
0answers
75 views

Series convergence Question ( TIFR GS $2010$)

Question is : What i have done so far is : I see that $\frac{\pi}{n}\rightarrow 0$ and so should be the sequence $u_n=\sin (\frac{\pi}{n})$. i.e., $u_n$ converges to $0$ and so $(b)$ is true. I ...
-1
votes
2answers
74 views

real-analysis Limit property question

$\text{Here is a question that I have been given, }$ $\lim_{x\to a}f(x)=A \iff $for each $c\in\mathbb R,\lim_{x\to a-c}f(x+c)=A$. For this question the proof follows: $\text{ Proof: Assume } ...
1
vote
1answer
128 views

A strange (seemingly pointless) exercise on convergence of series

I have come across an exercise which asks to prove that the series of functions$$\sum\frac{x^n}{1+x^n}$$ is convergent for $x\in [0,1)$. It also asks us to prove that the series converges uniformly ...
1
vote
0answers
79 views

Is the Convolution of a Schwartz Function with an $ L^{p} $-Function a Smooth $ L^{p} $-Function?

Let $ n \in \mathbb{N} $ and $ p \in \mathbb{R}_{\geq 1} $. If $ f \in \mathscr{S}(\mathbb{R}^{n}) $ and $ g \in {L^{p}}(\mathbb{R}^{n}) $, then it is a well-known fact from real analysis that the ...
1
vote
2answers
164 views

Mean value theorem and differentiability

Let $a,b\in \mathbb{R}$, $a<b$ and let $f$ be differentiable real-valued function on an open subset of $\mathbb{R}$ that contains $[a,b].$ Show that if $\lambda$ is any real number between $f'(a)$ ...
0
votes
1answer
23 views

Repeated transformation of function yields identity

For a function $f:\mathbb{R}\rightarrow\mathbb{R}$ in the Schwartz class, define $$Tf(y)=\dfrac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(x)e^{-ixy}dx$$ I want to show that $T^4f(y)=f(y)$. But plugging in the ...
1
vote
1answer
45 views

Is infimum achieved?

Suppose $V$ is a Banach space, and $V_0$ is a closed subspace of $V$. I know that the following quantity is well-defined for every $w\in V\sim V_0$: $$\inf\{\|w+v_0\|~|~v_0\in V_0\}$$ Is this infimum ...
3
votes
1answer
46 views

Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$

Let $f:[0,1]\to\mathbb{R}$ be continuous and $f(0)=f(1)$. Show that for any integer $n\geqslant 2$, there is $\xi\in(0,1)$ s.t.$f(\xi)=f\left(\xi+\frac{1}{n}\right).$ I think this requires the ...
0
votes
0answers
69 views

Prove that the set of injective linear transformations is an open set.

Prove that the set of injective linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^m$ is an open set. Using the fact that a Linear transformation is injective if and only if there is ...
2
votes
1answer
49 views

A dumb question on L'Hospital rule

In my book I've found the rule of L'Hospital rule as: If $f,g$ are diffentiable on $(c-\delta,c+\delta)-\{c\}$ and $g,g'\ne0$ on $(c-\delta,c+\delta)-\{c\}$ with $\lim_{x\to c}g(x)=\lim_{x\to ...
1
vote
2answers
104 views

Uniqueness of Fourier transform in $L^1$

The Fourier transform of an $L^1$ function is defined by $$\hat{f}(y)=\int_\mathbb{R}f(x)e^{-ixy}dx$$ Is it true that for functions $f,g\in L^1$, if $\hat{f}=\hat{g}$, then $f=g$?
0
votes
1answer
62 views

Prove x(t) is bounded given a integral inequality

I want to answer the following question: $x=x(t)$ is defined and continuous on $[0,T)$ and satisfies an integral inequality $$1 \leq x(t) \leq A_1 + A_2\int_0^t x(s)\big(1+\log x(s)\big) ds$$ for ...
4
votes
3answers
2k views

$\sqrt x$ is uniformly continuous

Prove that the function $\sqrt x$ is uniformly continuous on $\{x\in \mathbb{R} | x \ge 0\}$. To show uniformly continuity I must show for a given $\epsilon > 0$ there exists a $\delta>0$ such ...
2
votes
1answer
26 views

Integrating two variables for $L^1$ function

Suppose $f,g\in L^1(\mathbb{R})$. Is it necessarily true that $$\int_\mathbb{R}\int_\mathbb{R}|f(x-y)g(y)|dxdy=\int_\mathbb{R}|f(x)|dx\int_\mathbb{R}|g(y)|dy.$$
1
vote
0answers
11 views

represent some set to a functional form show example

Let say we have a set k $\subset R$ K = [a,b) for convenient k = $[0,\infty)$, How do we represent some function such as $f:K \to R$ can we represent f:k like y= ..... ?? or we can only ...
2
votes
2answers
53 views

limit of a sequence proof.

Let sequences $\{a_n\}$ and $\{b_n\}$ have the property that $\lim_{n \to \infty} [a_n^2+b_n^2]=0$. Prove that $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=0$. Proof: Suppose that $\lim_{n \to ...
1
vote
2answers
53 views

$f:[0,1]\rightarrow\mathbb{R}\in C[0,1]$ show $\lim_{a\rightarrow 0^+}(\int^1_a(t^{\frac{-1}{2}}f(t))\,dt)$ exists

$f:[0,1]\rightarrow\mathbb{R}\in C[0,1]$ (is continuous) Show that $$\lim_{a\rightarrow 0^+}(\int^1_a(t^{\frac{-1}{2}}f(t))\, dt)$$ exists I've thought about using integration by parts, but the ...
1
vote
1answer
120 views

Show $\sum^\infty_{n=1}(\frac{x}{n^{0.6}(1+nx^2)})$ converges uniformly on $\mathbb{R}$

$\sum^\infty_{n=1}\frac{x}{n^{0.6}(1+nx^2)}$ converges uniformly on $\mathbb{R}$ Is $x\rightarrow\sum^\infty_{n=1}(\frac{x}{n^{0.6}(1+nx^2)})$ continuous at all points of $\mathbb{R}$? I'm stuck on ...
1
vote
1answer
47 views

In a normed vector space $(V,\lvert . \rvert)$ show that $f:V\rightarrow \mathbb{R}$ with $f(v)=\lvert v\rvert$ is uniformly continuous

In a normed vector space $(V,\lvert . \rvert)$ show that $f:V\rightarrow \mathbb{R}$ with $f(v)=\lvert v\rvert$ is uniformly continuous The first part of the question says to prove the "reverse ...
0
votes
1answer
189 views

Quotient norm and actual norm

I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed. In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< ...
2
votes
1answer
45 views

Functions, Continuity and IVT

Suppose that $g$ is a function defined and continuous on $\mathbb{R}$ and $n$ is a positive integer such that $$\lim_{x\to \infty} \dfrac{g(x)}{x^n} = 0 = \lim_{x\to -\infty} \dfrac{g(x)}{x^n}$$ (i) ...
0
votes
1answer
103 views

about dual space

Im reading a chapter talking about dual space in optimization by vector space. I have captured something confusing as follow: I have been confused by two sentenses that one is " The mapping φ:X -> ...
0
votes
2answers
49 views

Functions and the IVT

Let $g, h$ be continuous functions defined on some interval $J$ and suppose that $g(x) \neq 0$ for any $x \in J$. If $g(x)^2 = h(x)^2$ for all $x \in J$, show that either $g(x) = h(x)$ for all $x \in ...
0
votes
1answer
35 views

On functions that are bounded by other certain functions

I am trying to address a specific, but also rather abstract, problem, which briefly can be stated as follows: Let $f$, $g$ be real-valued functions defined for all column vectors in $\Re^n$, i.e., ...
2
votes
2answers
366 views

prove that there exists a unique number x $\in$ R such that $f(x) =x$

The assumptions that are given in the statement $$f: R\mapsto R$$ and continuous and decreasing. me and my friend are doing this problem, we have attempted the following: $f(x) = x$, $c = 0$, c:= a ...
1
vote
2answers
38 views

Limit and maximum: IVT

Let $f$ be a function defined and continuous on $\mathbb{R}$. Assume that $f(a) > 0$ for some $a \in \mathbb{R}$ and that $$\lim_{x\to \infty} f(x) = 0 = \lim_{x\to -\infty}f(x)$$ Show that ...
0
votes
1answer
85 views

integration over unit ball

Can anyone give me some explanation about the following fact? Many thanks! Let $B$ be the unit ball in $\mathbb{R}^d$, $d\ge 1$. Then $$\int_{\mathbf{z}\in B}\frac{1}{|\mathbf{z}|^c}\,\mathrm{d} ...
2
votes
3answers
241 views

What do physicists mean with this bra-ket notation?

In Quantum mechanics we said that $\langle x'|\psi \rangle = \psi(x)$, where $\langle \phi|\psi \rangle $ is the dot product in $L^2(\mathbb{C})$. I found out, that this is true, if you set x' to ...
2
votes
0answers
60 views

uniform continuity and compact sets

I want to know what you think of this question. This might be does not make sense but I'm going to ask it anyway is it possible to ask? $$y= \sqrt{x} $$ is a compact set? Compact set ...
3
votes
1answer
133 views

If $x\le c\le y$ and $|x-y|<\delta$, then $|\frac{g(x)-g(y)}{x-y}-g'(c)|<\epsilon$

Alright, so here goes the question: Consider a function $g$ that is differentiable at some point $c$ on an interval $[a,b]$. Prove that: $\forall$$\epsilon$ $>0$ $\exists$$\delta$ $> ...
2
votes
2answers
64 views

$\sum_{m=0}^{\infty} \exp(-ma)\cos(mb)$

This is one of my homework problems: Let $a,b \in \mathbb{R}$ and $a>0$, show that: $$\sum_{m=0}^\infty \exp(-ma) \cos(mb)= \frac{1-\exp(-a)\cos b}{1- 2 \exp(-a) \cos b + \exp(-2a)} $$ I ...
1
vote
0answers
83 views

$ |\sin (x) | \leq 1$ by it's definition

I was told that it can be shown that $|\sin(x)| \leq 1$ by its definition $\sin (z):= \frac{1}{2i} \bigl( (\exp(iz)-\exp(-iz)\bigr) $ I am aware that as soon as I choose $x \in \mathbb{R}$ and ...