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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50 views

Let $f(z):=e^{\frac 1 z}, {z \in \mathbb C \setminus\{0\}}$.What values of $z$ are $ f(z)=re^{i\phi}$ for $r\in(0, \infty), \phi\in\mathbb R$?

Consider $f(z):=e^{\frac 1 z}, {z \in \mathbb C \setminus \{0\}}$. For which values are $f(z)$ real ? I've considered $e^{\frac 1 {a+ib}} = e^{\frac {a-ib} {a^2+b^2}}$. For which values are $f(z)$ ...
2
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1answer
62 views

Is this just asking for the Uniqueness of Limits? If not, how can you do this?

Suppose real function $f$ is continuous at every point. Prove that $f(x)=c$ cannot have two solutions for every value of $c$. I think you can prove this by contradiction assuming there are two ...
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1answer
212 views

measurable subset of nonmeasurable set

show that if E is measurable and E⊂P where P is nonmeasurable set in [0,1), then m(E)=0. Can one please tell how to start .. and I have one more question: is the union of m'ble set and non-m'ble set ...
0
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1answer
19 views

Increasing rate of a continuous function

Consider $f: X \rightarrow X$ continuous, with $X \subset \mathbb{R}^n$ compact convex. I am wondering on conditions on $f$ so that there exists $\epsilon > 0$ such that $$ (x-y)^\top \left( f(x) ...
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1answer
28 views

Sufficient condition for equality of two radon measures

Let $ X $ be a locally compact Hausdorff space and let $ \phi_1 $ and $ \phi_2 $ be two Radon measures on X (outer measure means measure and the definition of Radon measure that I am assuming can be ...
0
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1answer
34 views

How would Rolle's Theorem be used to prove this?

g is differentiable on (a,b) and for all x in [a,b] $a<g(x)<b$, and $|g'(x)|<1/2$. Prove $g(x)=x$ has at most one solution in [a,b]
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1answer
395 views

$\epsilon - N$ proof confirmation.

These proofs seem to be my absolute worst problem. I just don't seem to get them, that being said, if this is right, I may have started to get the hang of it. My limit and required assumptions: ...
0
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1answer
128 views

Why an integral does not exist?

I am trying to construct a counter example of Fubini Thorem, and for that we need a function $f$ in the product space which is not absolute integrable. So, let ...
3
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1answer
39 views

Showing $\lim_{n\to\infty} \left( \frac{n}{n^2+1^2} + \frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} \right) = \frac{\pi}{4}$

How could I go about proving the following limit: $$ \lim_{n\to\infty} \left( \frac{n}{n^2+1^2} + \frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} \right) = \frac{\pi}{4} $$
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1answer
40 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
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0answers
23 views

Computing area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions

I need to compute the area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions. As I am a non mathematics guy, it will be great if someone helps me out with the ...
1
vote
2answers
146 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
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1answer
59 views

Real Analysis question on sequences (Hint needed!!!)

Given the number $\alpha > 1$ , define the sequence an where $a_0 = 1$ and $a_{n+1} = (\alpha \times a_{n})^{\frac{1}{4}}$ for $ n \geq 0 $. Prove: If $a_{n}^{3}< \alpha $(as is true when n ...
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3answers
269 views

Take 2: When/Why are these equal?

This didn't go right the first time, so I'm going to drastically rephrase the query. As per this previous question, I am wondering if the two series ...
2
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1answer
89 views

Prove where $|x|^2(\sin(\pi|x|))^2$ (piecewise) is differentiable in $\mathbb{R}^2$

List all points in $\mathbb{R}^2$ at which $f$ is differentiable as well as ALL points in $\mathbb{R}^2$ where $f$ is not differentiable (implied by the first list) when \begin{equation} f(x) = ...
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1answer
64 views

Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
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1answer
28 views

Proving that sng(x) is discontinuous at 0?

I am trying to prove that the signum function is discontinuous at $x_0 = 0$. The criterion for discontinuity is that if there is a sequence $(x_n) \subseteq A$ and $(x_n) \rightarrow c$ but $f(x_n)$ ...
3
votes
1answer
36 views

When are these series equal?

Suppose we have a power series $$\sum_{n=0}^\infty {a_nb_nx^n}$$ When is it true that the series obtained by eliminating $b_n$ is proportional to the original series? $$\sum_{n=0}^\infty ...
5
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2answers
176 views

Subsets of $[0,1]$

Suppose we have a closed subset $A\subset[0,1]$ that is not equal to $[0,1]$. Is it possible $mA=1$? Suppose you have an open subset $B\subset[0,1]$ that is dense in $[0,1]$. Is it possible that ...
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3answers
42 views

How do you evaluate $f(x_n)$?

I am not sure I am clear on how to evaluate the $f(x_n)$; what the it mean to evaluate the function of a sequence?
2
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1answer
35 views

$(\epsilon,N)$ proof that $\displaystyle{\lim_{n \to \infty} \frac{n^3-3n^2+4n-7}{2n^3-10n^2+1} = \frac{1}{2}}$

I need to find an $N \in \mathbb{N}$ such that: for all $\epsilon>0$, there exists $N>0$ such that for all $n \geq N$ $$\left|\frac{n^3-3n^2+4n-7}{2n^3-10n^2+1} - \frac{1}{2} \right| < ...
1
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1answer
58 views

integral $I=\int_{-\infty}^\infty e^{-\alpha x^{2k}}dx$

$$ I=\int_{-\infty}^\infty e^{-\alpha x^{2k}} dx $$ The last problem was ill posed, and is answered in the post! You can disregard this post!
14
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1answer
413 views

Integral $\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
0
votes
1answer
19 views

Question about outer regularity and inf

Let $\mu$ be a measure. Suppose for every $\varepsilon > 0$, there exists an open set $U \supset E$ such that $\mu(U) < \mu(E) + \varepsilon$. Then must $\mu(E) = \inf\{\mu(U): U \supset E, U ...
0
votes
0answers
71 views

Proof by contradiction and order of statements

Order matters, take sequences tending towards a limit: $$\forall\epsilon>0\exists N\in\mathbb{N}:n>N\implies|x_n-L|<\epsilon$$ "For all $\epsilon$ there exists an $N$" is totally different to ...
2
votes
1answer
63 views

Showing that if derivative is 0, function is constant ($f: U \rightarrow \mathbb{R}$ where $U \subset \mathbb{R}^n$)

Here's the question: Suppose that $f: U \rightarrow \mathbb{R}$ is differentiable on the open subset $U\subset \mathbb{R}^n$, and $Df(x) =0$ for all $x\in U$. Show that $f$ is constant on $U$. My ...
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1answer
48 views

Error in proof of extreme value theorem (Protter and Morrey's book)

The bounded theorem has just been proved and thus is/may be used. $f:[u,v]\rightarrow\mathbb{R}$ and is continuous (so it is bounded) We wish to show $\exists a,b\in[u,v]:f(a)=m, f(b)=M$ where ...
0
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2answers
197 views

Show where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. My thoughts: For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = ...
1
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1answer
76 views

Jordan decomposition of sum of two measures

Let $\mu$ and $\nu$ be finite signed measures. Then by the Jordan Decomposition Theorem, we can write $\mu = \mu^{+} - \mu^{-}$ and $\nu = \nu^{+} - \nu^{-}$ where $\mu^{\pm}, \nu^{\pm}$ are unsigned ...
4
votes
1answer
90 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
-1
votes
2answers
50 views

Epsilon delta proof for the following limit

How do write an epsilon-delta proof or the following limit assuming the limit is -2, $$\lim_{x \to -1} \frac{x^2-1}{x+1}$$
2
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2answers
139 views

Uniform convergence of Lagrange polynomials

There is a well-known theorem that states that on a closed interval $[a,b]$ any continuous function is the limit of a uniformly convergent sequence of polynomials. Proofs for this theorem usually ...
0
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1answer
154 views

Mollification of a function continuous on $\mathbb{R}\backslash\{0\}$, need uniform convergence

We know that if $f:\mathbb{R} \to \mathbb{R}$ is continuous, then its mollification $f_\epsilon$ converges uniformly to $f$ on compact subsets of $\mathbb{R}$ as $\epsilon \to 0$. My question is, ...
1
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1answer
30 views

$\displaystyle \lim_{n\to \infty}{(\int_{0}^{1}{(f(t))^ndt})^{\frac{1}{n}}}=\sup_{x\in[0,1]}{f(x)}$ for Riemann integrals without using Lebesgue

I know how to prove this result using lebesgue integral. But I want to know if someone knows a proof that works for Riemann integral. I know that we can translate the problem into a Lebesgue Integral ...
0
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1answer
22 views

Redefine a discrete compact set

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,\ldots,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as ...
1
vote
1answer
88 views

$x_n,x_ny_n$ convergence implies $y_n$ converges

Assume that $x_n$ converges to a nonzero number $x$ and that the sum $x_ny_n$ converges to a limit $L$. Prove that the series $y_n$ converges. The natural guess is that $y_n$ will converge to $L/x$. ...
1
vote
1answer
83 views

Prove that $\,f(z) \equiv 0,\,$ if $\,f(z)\,$ is entire

Let $f(z)$ be an entire analytic function, such that $$ \int_{0}^{2\pi}\lvert\, f(re^{i\theta}) \rvert\,d\theta\le r^{16/5}, \quad \text{for all}\,\,\, r>0 $$ Show that $$f(z)\equiv 0.$$ Thank ...
0
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0answers
61 views

A basic question on limit calculation [duplicate]

How to prove that the following limit exist without calculating its value $$ \lim_{t \to\infty} \int_{0}^{t}\frac{\sin x}{x} dx $$
0
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1answer
40 views

Determine$\int_{a}^{b}f'(x)(f(x))^s)dx,s\in \mathbb Z$

Let $f:[a,b] \rightarrow \mathbb R$ be continuously differentable and $f(x)>0 , x\in [a,b]$. Determine the following integral: $(1)\int_{a}^{b}f'(x)(f(x))^s)dx,s\in \mathbb Z$ At ...
2
votes
2answers
71 views

If $f$ and $g$ are continuous on $[a,b]$, $f(a) \le g(a)$, and $g(b) \le f(b)$, prove there is a c in [a,b] with $f(c)$ $=$ $g(c)$.

If $f$ and $g$ are continuous on $[a,b]$, $f(a)$ $\le$ $g(a)$, and $g(b)$ $\le$ $f(b)$, prove there is a point $c$ $\in [a,b]$ such that $f(c)$ $=$ $g(c)$. Any ideas on how to solve? I think I have ...
2
votes
1answer
82 views

Is the dual of a complete topological vector space always complete?

Let $X$ be a complete topological vector space (over $\mathbb{C}$ say), and $X'$ its dual with the weak*-topology. Then is $X'$ always complete? You may assume $X$ is locally convex if you like.
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1answer
47 views

How is the Mean Value Theorem used when dealing with second derivative in this question?

Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and twice differentiable on $(a,b)$. The line segment joining points $(a,f(a))$ and $(b,f(b))$ meets $f$ at $c$,which lies between $a$ and $b$. ...
0
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2answers
60 views

arc wise connected set

I am having confusion in understanding what is arc wise connected set.The definition is a set $S$ is arc wise connected if for any pair of point a,b we can define a continuous function $f$ from ...
0
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1answer
69 views

Double Integral Proof

Let function $f(x, y)$ be defined by $$f(x, y) =\begin{cases} 1,\text{ if }x = y,\\ 0,\text{ otherwise}.\end{cases}$$ Using the definition of the double integral show that the following integral exists ...
0
votes
1answer
26 views

Define a compact and convex set through inequality constraints

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x ...
0
votes
2answers
83 views

A basic question on rational and irrational number

Suppose we take all the rationals and take any neighbourhood around each of them. Will they cover whole $\Bbb R$. I think so as rationals are dense. So, for each irrational we can find a rational ...
2
votes
1answer
86 views

Weierstrass function

I got stuck on this exercise from Prof. Tao's real analysis notes. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic ...
4
votes
3answers
58 views

Help verifying my proof that for any $\epsilon>0$, there exists rational $\frac{j}{k}$ such that $0<\frac{j}{k}<\epsilon$

I'm trying to prove that that for any $\epsilon>0$, there exists rational $\frac{j}{k}$ such that $0<\frac{j}{k}<\epsilon$. Obviously $j,k\in \mathbb{N}$. This is not for homework, it's a ...
2
votes
1answer
31 views

$\mathbb R$ as union of nowhere dense set and null set.

It is well known that $\mathbb R$ can be written as a union of a null set and a meagre set (set of first category). Can $\mathbb R$ also be written as a union of a null set and a nowhere dense set ? ...
-1
votes
2answers
79 views

Proving a limit Epsilon Delta definition

How do I find and prove the limit of the following function using the epsilon delta definition. $$\lim_{x \to 0} \frac{2+\sin x}{3-\cos x}x$$