Questions tagged [real-analysis]
For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.
145,638
questions
2
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2
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Is Lebesgue density independent of the choice of neighborhoods?
Let $E$ be a Borel subset of $\mathbb R^d$ and $m$ be the Lebesgue measure on $\mathbb R^d$ and by definition of density (Folland real analysis 2nd edition ,Exercise 25 in page 100):
$$D_E(x):= \lim_{...
- 7,939
0
votes
1
answer
96
views
Proving a function continuous at $0$.
Define for $-\pi<\arg z<\pi$ $$f(z)=\frac{1}{1-\sqrt{z}}$$ and define $$f(0)=1$$
Question Prove that $f(z)$ is continuous at $0$.
My try- Define $$g(z)=\sqrt{z}$$
Define $g(0)=0$.
Let, $\epsilon&...
- 19
2
votes
1
answer
101
views
Asymptotic Estimation of a function ${\int_{0}^{x} \frac{\sin^2 t}{t^2} dt}$
I'm reading a book "Theory of Approximation of Functions of a Real Variable" by A.F Timan
On the page 14 there is an asymptotic estimation:"... It is an integral function of the first ...
- 708
2
votes
0
answers
69
views
Confusion regarding solution manual's entry to Question 3b Chapter 8 of Spivak's Calculus
Question 3b from Chapter 8 of Spivak's Calculus reads as:
The proof of the Theorem 7-1 depended upon considering $A = \{x: a \leq x \leq b \text{ and } f \text{ is negative on } [a,x]\}$. Give ...
- 4,984
0
votes
1
answer
21
views
Bounds of double integrals determined by a minimum
I'm studying for exams and in the previous years there has been a question on a double integral where the domain has the condition : $ |y| <= min(1,2-x) $
My problem is that I don't understand how ...
- 5
1
vote
2
answers
365
views
The simplest ODE $x' = x$
An ODE $x'(t) = x(t)$ is one of the simplest equation.
If $x(t) \in C^1(\mathbb{R})$ is a solution of this ODE, we get $x(t) = C \exp(t)$, where $C$ is constant. This is very elementary fact in ODE ...
- 627
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votes
1
answer
276
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Finding Limit vs Proving Limit of *something*.
I get confused everytime between finding a limit and proving that something is a limit. If I calculate the limit of a function $f(x)$ at $a$ and say, I get $\lim\limits_{x \to a} f(x) =L$ where $a\in [...
- 4,893
2
votes
4
answers
2k
views
Prove that $|x|<\epsilon$ for every $\epsilon$ greater than $0$ if and only if $x$ is equal to zero.
Prove that $|x|<\epsilon$ for every $\epsilon$ greater than $0$ if and only if $x$ is equal to zero.
My Attempt:
Assume $x$ to be a non-zero number, say $x=2$. Clearly there is a contradiction here ...
- 9,220
0
votes
0
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42
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9.8 Mathematical Analysis by Apostol
Can someone please explain to me the 2nd point in question below?
In point 2 what does the condition $|f_k(x)-f(x)|<\delta$ means, is it true for all $k$ or does this $k$ depends on $x$?
- 91
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1
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238
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If $a_n=\sqrt{1+\sqrt{2+\cdots\sqrt{n}}}$ and $\lim\limits_{n\to\infty}a_n=\ell$, prove $\lim_{n\to\infty}[(\ell-a_n)^{1/n}n^{1/2}]=\frac{\sqrt e}2$
$$a_n=\sqrt{1+\sqrt{2+\cdots\sqrt{n}}}$$
We can prove that $\{a_n\}$ is convergent
(using mathematical induction, $\sqrt {k+\sqrt{k+1+\cdots\sqrt{n}}}\leq k-1, for \ k\geq3$).
If
$$
\lim\limits_{n\to\...
- 773
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votes
2
answers
744
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If the flux of a vector field $F$ is zero for ANY surface $S$ then is $F$ zero?
If $\vec F$ is a vector field defined in a open (contractible) set $U$ of $\Bbb R^3$ such that
$$
\int_S\vec F\cdot\hat ndS=0
$$
for ANY surface $S$ then in particular the last equality holds for the ...
- 4,832
0
votes
1
answer
51
views
Rigorous proof of existence of a unique function whose arc-length is minimal.
Let $a,b, \alpha, \beta \in \Bbb R$, $F = \{ f \in C^1([a,b]) \mid f(a)=\alpha, f(b)=\beta\}$ and $L : F \to \Bbb R^+, f \mapsto \int_{a}^b \sqrt{1+f'(x)^2} dx$.
Prove that $L$ has a unique minimum.
...
- 1,784
2
votes
0
answers
54
views
Show that $\lim\limits_{t \to 0}\frac{Det(A+tB)-Det(A)}{t} = Det(A)\cdot Trace(A^{-1}B)$
Show that $\lim\limits_{t \to 0}\frac{Det(A+tB)-Det(A)}{t} = Det(A)\cdot Trace(A^{-1}B)$.
Also show that $\langle \nabla Det(A),B \rangle=Det(A)\cdot Trace(A^{-1}B)$.
I know that $det(tI_{n\times n}+C)...
- 67
0
votes
1
answer
53
views
Integral remainder from Taylor's formula
The wikipedia version of the integral remainder from Taylor's theorem says (if I want an order 2 expansion)$$f(y)=f(x)+f'(x)(y-x)+{1\over2}f''(x)(y-x)^2+{1\over2}\int\limits_x^y(y-z)^2f'''(z)dz$$
But ...
- 979
0
votes
2
answers
116
views
Can any $f:\mathbb N \to \mathbb N$ be differentiable at a point?
I know that any function on $\mathbb N$ is continuous but is any function differentiable on $\mathbb N$? For example if $f(x) = x^2, f: \mathbb N \to \mathbb N$ and let $c = 2$, can we claim the ...
- 478
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0
answers
40
views
Let $f: \Bbb{R^n} \rightarrow \Bbb{R}$ be a function such that $\forall q \in \Bbb{Q}$, the sublevel set $\{x: f(x)<q\}$ is open...
This is a question in old qualification exam. Given the condition mentioned by the title, I am asked:
a) Must $f$ be Borel-measurable?
b) Must $f$ be continuous?
First, I know that b) is true since ...
- 1,338
0
votes
0
answers
270
views
Limit of the convolution of two functions as the product of the integral of one and the limit of the other
Let $\lambda, h : \mathbb{R} \to \mathbb{R}$ and suppose $\int_{-\infty}^\infty h(t)\,dt$ exists and $\lim_{t \to \infty}\lambda(t)$ exists. I want to show that the limit of their convolution equals ...
- 43
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votes
0
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97
views
Does differentiating a differentiable function give a differentiable function?
I want to know that if a differentiable function is differentiated, it is still differentiable.
I mean, let $f \ $ be a differentiable function, and $f' \ $ its derivative.
Is $f' \ $ always also a ...
- 71
0
votes
2
answers
329
views
metric space without isolated points
Let $X$ be a compact metric space with no isolated points (so every point is a limit point). Show that $X$ is uncountable.
I think I can show this by showing that $X$ has cardinality at least that of ...
- 859
0
votes
1
answer
64
views
Inequality with a matrix norm
Let $T\in \mathcal L(\mathbb R^n, \mathbb R^m)$ and $A = [a_{ij}] \in \mathcal M_{m\times n}$ the associated matrix of this transformation. If $\mu := \max \{ |a_{ij}| : i =1, \dots, m; j = 1, \dots, ...
- 551
3
votes
3
answers
664
views
connected and path components
Let $F = G\cup H$ where $G = \{(x,y) \in \mathbb{R}^2 : x=0, y = [0,1]\cap \mathbb{Q}\}$ and $H = \{(x,y) \in \mathbb{R}^2 : x > 0, y \in [0,1]\backslash \mathbb{Q}$. Find the connected components ...
- 859
1
vote
0
answers
40
views
Perpendicular continuous function of $R^3$ unit vectors?
Let S be the set of all unit vectors in $\mathbb{R}^3$
Does there exist a continuous function $f$ from $S \to S$ such that $\forall v \in S: v \cdot f(v) = 0 $ ?
If so, what is one such function? If ...
- 2,650
0
votes
0
answers
35
views
Finding certain improper integrals
I need to find two functions,
$f, g : (0,1] \rightarrow \mathbb{R} $, such that both integrals $\int_{0}^{1} f(x) dx $ and $\int_{0}^{1} g(x) dx $ are improper and exist but the product is an ...
- 537
1
vote
1
answer
395
views
metric spaces with certain topological properties
Define for a field $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}, \mathbb{F}^\omega := \{(x_n)_{n\geq 1} : x_i \in \mathbb{F}\,\forall i\}.$ Let $x=(x_n)_{n\geq 1}, y=(y_n)_{n\geq 1} \in \mathbb{R}^\omega$....
- 1,146
23
votes
1
answer
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Delightful integral: $\int _0^{\frac{\pi }{2}}x\ln \left(\sin \left(x\right)\right)\ln ^2\left(\cos \left(x\right)\right)\:dx.$
A friend of mine proposed me this integral which I find to be very interesting.
I managed to find with the help of software that:
$$\int _0^{\frac{\pi }{2}}x\ln \left(\sin \left(x\right)\right)\ln ^2\...
- 1,111
1
vote
1
answer
73
views
Determine the convergence of following series
Determine the convergence of following series.
$$ \sum_{n=1}^\infty a_n$$ where $$ a_n =
\begin{cases}
\bigg( \dfrac{2+3n}{5+6n} \bigg)^n, & \text{if $n$ is even} \\
5 & \text{if $n$ is ...
- 357
3
votes
0
answers
95
views
Baby Rudin 1.20(b) Proof—Alternative Justification
This question is based on a previously asked one that I will quote in full here:
I have a question about Rudin's proof of Theorem 1.20 (b) in his book Principles of Mathematical Analysis. Theorem 1....
- 19.7k
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votes
0
answers
65
views
How does the standard topology on $R$ induce a structure on $R$?
I'm having a hard time seeing how does the standard topology on $R$ induces the structure on the set of real numbers such that it give us the real number line! The definition of a topology on a set ...
- 569
0
votes
3
answers
341
views
Definition of Sequence Convergence Doesn't Make Sense
In real analysis, the convergence of a sequence is defined as follows:
The statement that $|a_n - a|<\varepsilon$ means that the range is $[a-\varepsilon, a+\varepsilon]$, where $\varepsilon>0$....
- 11
0
votes
0
answers
63
views
Given $ f: [a, b] \to \mathbb R $ and $ n \in \mathbb N $, assume $ h = \frac {b-a}{}\cdots$
Given $ f: [a, b] \to \mathbb R $ and $ n \in \mathbb N $, assume $ h = \frac {b-a} {n} $ and the average of the numbers $ f ( a + h ), \ldots, f (a + nh) = f (b) $ given by
$$ M (f; n) = \frac {1}{n} ...
- 69
2
votes
0
answers
34
views
if $f$ is continuous over some interval $[z_1,z_2]$, then $f(bx+c)$ is continuous over the adjusted interval $[\frac{z_1-c}{b}, \frac{z_2-c}{b}]$
I want to prove that if $f$ is continuous over some interval $[z_1,z_2]$, then any horizontally translated and horizontally "compressed/stretched" version of $f$ (i.e. a function of the form ...
- 4,984
1
vote
1
answer
408
views
Conjugate Gradient and Krylov Subspace Method
In HF, on each iteration the CG algorithm is used to approximately compute
$$
\mathbf{d}=-(\mathbf{H}+\lambda \mathbf{I})^{-1} \mathbf{g}
$$
where $\mathrm{d}$ is the step direction, and $\mathrm{g}$ ...
- 1,505
1
vote
2
answers
88
views
$f(X)\subset \{0\}\cup f(\text{supp}(f))$ vs $f(X)\subset f(\text{supp}(f))$
Suppose $f$ has compact support and is continuous (say we are in the simplest case of $f:\mathbb R\rightarrow \mathbb R$).
I was reading this answer, and can't understand why there needs to be a $\{0\}...
- 979
1
vote
0
answers
50
views
Bounding the total variation of an Absolute Continuous function
Theorem: Suppose $f \in AC([a,b])$. Prove that
$$
V_a^b(f) \leq ||(f')^+||_{L^1([a,b])} + ||(f')^-||_{L^1([a,b])}
$$
Proof (my attempt): If $f \in AC([a,b])$ then calculus formula holds and we can ...
- 58
0
votes
1
answer
57
views
Proof that $x^2$ is continuous by $x = x_0 +h$ definition
Let's first sketch the proof:
$$|f(x_0 + h) - f(x_0)| = |(x_0 +h)^2 - x_0^2| = |h^2 + 2x_0h| \leq|h||h + 2x_0|...\text{Now let's say that |h|$< \delta$ $...$}\\ |h||h + 2x_0|< |h||1 + 2x_0| <...
- 2,527
3
votes
3
answers
125
views
Let $a_n$ be a sequence of nonnegative numbers such that $a_n \rightarrow b$. Then, $\lim_{n \rightarrow \infty} \frac{(a_{n+1})^{n+1}}{(a_n)^n}$ =?
As the title says, I would like to evaluate the following limit given that $a_n \rightarrow b < \infty$: $\lim_{n \rightarrow \infty} \frac{(a_{n+1})^{n+1}}{(a_n)^n}$ (I am assuming that the limit ...
- 1,338
1
vote
0
answers
161
views
Approximation for log of convex combination
Thinking about the approximation of
$$
\ln \sum_{i=1}^{N}\theta_{i}x_{i},
$$
where $\{\theta_i\}$ are positive constant that satisfies $\sum_{i=1}^{N}\theta_i= 1$ (i.e., weights) and $\{x_{i}\}$ are ...
- 11
2
votes
2
answers
3k
views
Different definition for L smooth function.
At optimization class, professor gave the definition of L smooth function by
$f:\mathbf{R^n} \rightarrow \mathbf{R} $ is L smooth if all the eigenvalue for $\nabla^{2} f $ is smaller than L
where $\...
- 41
0
votes
0
answers
55
views
Continuity of finite intersection of continuous correspondence
I was solving a problem which requires me to show the continuity of a set-valued map. Suppose $\Bigl(T(x_i)\Bigr)_{i=1}^{n} \subset A$, where the set-valued map $T$ defined as $\mathbb{X} \ni x \...
- 41
4
votes
2
answers
1k
views
If $f$ is strictly increasing on $[a,b]$ then $f$ is continuous at some point in the interval.
Edit: Proof version 2.0
Thanks to all for your insight and suggestions.
Theorem: If a function $f$ is strictly increasing on the closed interval $[a,b]$, then $f$ is continuous at some point in the ...
- 1,585
2
votes
0
answers
67
views
Attempt at prove that rank theorem implies that the image of this function with constant rank is a manifold
Let $f:U\rightarrow U$, $U\subseteq\mathbb{R^m}$ open, such that $f\circ f = f$ and let $M=f(U)$. I showed that there exists neighborhood $V\subseteq U$ of M such that $f$ has constant rank on $V$, ...
- 588
1
vote
2
answers
333
views
Inverse Function Theorem and finding local inverses
$f(x,y) = (x+y,x^2+y)$ for this function, the question is Write down a $C^1$
local inverse around (0, 0).
I wonder how can I find the points f has a local inverse? I am trying to use Lagrange ...
- 43
1
vote
0
answers
44
views
If $q(t)\underset{t\to +\infty}{\to}1$ then number of zeros of solution of $x^{''}+q(t)x=0$ verify $N(t)\underset{t\to +\infty}{\sim}\frac{t}{\pi}$
Let $q:\mathbb{R}^+\to \mathbb{R}$ continuous function such that $q(t)\underset{t\to +\infty}{\to}1$. We consider $\phi$ solution of differential equation : $x^{''}+q(t)x=0$ and $N(t)$ is the number ...
- 61
2
votes
1
answer
231
views
Equivalence of definitions of the Axiom of Continuity in Economics
Statement $1$: If $(a_n, b_n)_{n \geq 1}^{\infty}$ is a sequence in $X \times X$ satisfying $a_n \succsim b_n$ and $\displaystyle\lim_{n \to \infty} (a_n, b_n) = (a,b)$, then $a \succsim b$.
Statement ...
- 585
1
vote
1
answer
74
views
Finding the interior and boundary of a subset of $(l^1,||\cdot||_{1})$
Consider the folllowing metric space: $X = (l^1,||\cdot||_{1})$. Recall $$l^1 = \big\{\{x_k\}_{k=1}^{\infty}: \sum_{k=1}^{\infty}|x_k| < \infty\big\}.$$
Let $Y \subset X$ be the set of sequences $\{...
user715112
0
votes
1
answer
286
views
Uniform and pointwise convergence of $x^n$ on $[0,1]$
Show that the sequence of functions $f_n(x) = x^n$ defined on $[0, 1]$ converges pointwise but not
uniformly. Now suppose $g \in C([0, 1])$ such that $g(1) = 0$. Show that the sequence $f_ng$ ...
- 43
1
vote
1
answer
124
views
Inequalities for limit supremum
We know that for any bounded sequences $(s_n)$ and $(t_n)$ the inequality holds $\limsup (s_n+t_n) \leq \limsup (s_n) + \limsup (t_n)$. I understand how to prove this inequality, but why doesn't this ...
2
votes
1
answer
513
views
Question regarding the fourier series, the L2 space of square integrable functions and convergence.
I have been self learning the fourier series and if anyone can help me with my questions that will be super great!
Firstly, why is the fourier series defined on the L2 space and not on other spaces? ...
- 153
0
votes
0
answers
38
views
Global minimum and local minimum question and proof of $x^2$ local minimum at $x = 0$
We defined $f : [a,b] \rightarrow \mathbb{R}$ has a global maximum in $x_0 \in [a,b]$ on $[a,b]$ if $x \neq 0 \implies f(x) < f(x_0)$
Then we also defined local maximum:
$f : [a,b] \rightarrow \...
- 2,527
2
votes
1
answer
309
views
Proof verification of theorem regarding $m$-tail of a sequence
Prove that a sequence $X=(x_n)$ converges iff its $m$-tail $X_m$($m\in \mathbb N$) converges. For that case, $lim X=lim X_m$.
Proof:If $X$ converges to $x$, then for a given $\varepsilon>0$, $\...
