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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16 views

continuous function of the approximation of a sequence 2

A followup on my previous question, does the following hold? Should make things simpler for powers instead of just addition. If $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = 1$ and $\lim_{n ...
0
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1answer
47 views

A counterexample for Lebesgue's Dominant Convergence theorem - where is my mistake?

I am having some trouble with Lebesgue's Dominant Convergence theorem. It seems as if I have a counterexample, and I can't find my mistake. Say that $\mu$ is a uniform measure over ...
1
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1answer
23 views

Convergence of $\sum_{n=2}^{\infty} a^{\log_en}$

The series $$\sum_{n=2}^{\infty} a^{\log_en}$$ converges for what values of $a$? Attempt: I am trying to use the Raabe's test / Logarithmic test . Let $u_n = a^{\log_en} \implies \dfrac ...
2
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1answer
71 views

Example 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How to show this set to be open?

Let $N$ be the following subset of $\mathbb{R}^2$: $$N \colon= \{ \ (x,y) \in \mathbb{R}^2 \ \colon \ \vert x \vert < \frac{1}{y^2+1} \ \}.$$ Then intuitively it is apparent that $N$ is open. ...
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1answer
51 views

Uniform continuity preserves uniqueness of convergent sequences?

If $f: (a, b) \to \Bbb R$ is uniformly continuous, $\{x_n\}$ and $\{x'_n\}$ are sequences in $(a, b)$ with $x_n \to b$, $x'_n \to b$, $f(x_n) \to y$, and $f(x'_n) \to \overline{y}$, prove $y = y'$. ...
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2answers
42 views

How to Prove that $\frac{1}{n}\sum_{k=1}^{n}a_k\rightarrow0 $ iff $\frac{1}{n}\sum_{k=1}^{n}a_k^2\rightarrow0 $.

Let $(a_n)$ be a sequence such that each $a_n\in (0,1)$. prove that $\frac{1}{n}\sum_{k=1}^{n}a_k\rightarrow0 $ iff $\frac{1}{n}\sum_{k=1}^{n}a_k^2\rightarrow0 $. I don't have a clear idea about ...
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0answers
70 views

root interlacing condition for this problem?

Consider the quasi-polynomial $$ F(z)=(a_3z^3+a_2z^2+a_1z+a_0)e^{\tau z}+(b_2z^2+b_1z+b_0)\;, $$ where $a_j,b_j\in\mathbb{R}$ and $\tau\in\mathbb{R}^{+}$. One can decompose $F(z)$ into even and odd ...
1
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1answer
71 views

Type of discontinuities in Dirichlet function

I notice three types of discontinuities "removable", "jump", "infinite" defined here Classification of Discontinuities involve limit. Then I am puzzled that what is the type of discontinuities in the ...
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4answers
88 views

Easy proof of $\mathcal{P}(\mathbb{Q})$ is uncountable [Big list]

I'm looking for a easy proof of uncountability of $\mathcal P(\mathbb Q)$. I'll contribute with this: Let $\mathcal{P}(A)$ denote the power set of $A$, since ...
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0answers
45 views

Proving that the given function is not piecewise continuous.

Let $f(x) = x\,\text{sgn}\left({\sin(\frac{1}{x})}\right)$where sgn is the signum function.. For $x\neq0$ and $f(0) = 0$ , how to prove that the given function is not piecewise continuous on $[-1,1]$. ...
2
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2answers
66 views

Let $f$ be continuous on $I=[a,b]$ such that $f(a)<0$, $f(b)>0$, $W= \{x\in I: f(x)<0\}$, $w=\sup W$ Prove that $f(w)=0$

Let $f$ be continuous on $I=[a,b]$ such that $f(a)<0, f(b)>0, W= \{x\in I: f(x)<0\}$, $w=\sup W$ Prove that $f(w)=0$ [I can see that this is an alternate proof for the Location of roots ...
3
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1answer
81 views

A possible inequality related to binomial theorem (or, convex/concave functions)

Let $x, \ y, \ p$ be any real numbers with $x>0$, $y>0$, and $p>1$. The question is about (most probably) an elementary inequality: Is it always true that $x^p+y^p\leq (x+y)^p$ ? Note ...
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1answer
133 views

Prove/disprove space is complete with metric defined by an integral (triangle inequality still missing in metric part)

I have a two part question: I need to show that $d(f,g)=\int_{-1}^1\! |f(x)-g(x)| \, \mathrm{d}x$ is a metric in $C((-1,1),\mathbb{R)}$ and furthermore prove/disprove that the space ...
2
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1answer
28 views

Equivalence of sequence spaces

Let $m$ be the space of infinite sequences $(x_i), |x_i| \leq 1$ with norm $\sup_{i>0}|x_i|$. Let $\ell$ be the space of infinite sequences $(x_i), \sum_{i> 0}|x_i| \leq 1$ with norm $\sum_{i ...
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1answer
48 views

Volume integration over a region

If $E_1$ is a region in the first octant that is bounded by the spheres $x^2+y^2+z^2=4$, $x^2+y^2+z^2=9$ and the plane $y=\frac{x}{\sqrt{3}}$, what is $$\int\int\int_{E_1} xdV?$$ I don't really have ...
3
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5answers
149 views

How to compute this multivariable limit?

How do I evaluate $$\lim_{x \to 0 ,\, y \to 0} \frac{x^3y-xy^3}{(x^2+y^2)^{3/2}}$$ I tried using squeeze theorem and writing it in polar coordinates, but I got stuck. Can anyone give me a hint?
2
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1answer
58 views

How to show $ \sum_{n=1}^{\infty} (\sqrt{b_{n}}- \sqrt{b_{n+1}})$ converges?

Let $a_{n} \ge 0 \hspace{1cm} \forall n \in$ $ \mathbb{N} \cup \{0\}$. and $ \sum_{n=1}^{\infty} a_{n}$ converges and $ b_{n}=\sum_{k=n}^{\infty} a_{k} $ Then we have to prove that$ ...
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2answers
55 views

Find all $\alpha$ such that for any $x>-1$ we have $\ln(1+x)\leq x-\frac{x^2}{2}+\alpha x^3$

Here's a small problem I'm trying to solve: Find all $\alpha$ such that for any $x>-1$ we have $\ln(1+x)\leq x-\frac{x^2}{2}+\alpha x^3$ After moving some things to the left side we have: ...
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1answer
37 views

$\lim_{k \rightarrow \infty} \mu \left(\bigcup_{n=1}^k A_n \right) = \mu \left(\bigcup_{n=1}^\infty A_n \right)$

The problem I'm working on is: Prove that for a family of measurable sets $A_k$ in $[a,b]$ the following is true $$\lim_{k \rightarrow \infty} \mu \left(\bigcup_{n=1}^k A_n \right) = \mu ...
2
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0answers
25 views

Why do we make $E \subset X$ in the definition of function limit / continuity?

In Baby Rudin, in the definitions of continuity, we have metric spaces $X, Y$, a set $E \subset X$, and a function $f : E \to Y$. My question is what is the purpose of this subset $E$? Why don't we ...
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1answer
28 views

Uniformly Continous $\delta - \epsilon$ proof

Let f and g be uniformly continuous on an interval S. Show that fg is uniformly continuous on S if S is closed. I am having trouble understanding skipped steps in the proof: Assume S is closed. ...
2
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1answer
199 views

Can we construct a function that has uncountable many jump discontinuities?

I know that Dirichlet function has uncountable many discontinuities. I think they are removable, because the discontinuities can be removed by redefining the function values of the rational numbers as ...
2
votes
1answer
43 views

Basic Application of Implicit Function Theorem

I am studying an old exam for a course in real analysis and came across this problem (not a homework problem). Let $f(x, y, z) = (xyz, (z \cos x) + y - 1)$ and observe that $f(2\pi, 1, 0) = (0, 0)$. ...
3
votes
1answer
201 views

Convergence to the Dirac Delta Function

Let $h\colon[0,1]\to \mathbb{R}^+$ be any bounded measurable non-negative function with a unique maximum at $a$ and $h$ is continuous at $a$. For $\lambda>0$ define $h_\lambda(x)=C_\lambda ...
3
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1answer
59 views

Is this type of smooth function analytic?

Let $f:(a,b)\to\mathbb{C}$ be a $C^{\infty}$ application, such that for any $t_0\in (a,b)$ there is $g:(a,b)\to\mathbb{C}$, $g\in C^{\infty}((a,b)),\ g(t_0)\neq 0$ and $m\in\mathbb{N}^*$ such that: ...
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2answers
38 views

Why infinite expansions of basic functions and polynomial expansions use factorial in the denominator

I'm looking for a very basic applicable example of why in infinite series and other infinite expansions of basic/transcendental functions ($\log{x}$, $e^x$, $\sin{x}$, power series, etc) use ...
3
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0answers
69 views

Does a Plancherel-style theorem for the Hardy space $\mathcal{H}^2(\mathbb{T})$ exist?

I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathcal{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I ...
2
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0answers
60 views

Prove the part of the graph of $f$ on $I$ is never below the tangent line to the graph at $(c,f(c))$.

Let $f''(x) \geq 0, \forall x\in I$. If $c\in I$, show that the part of the graph of $f$ on $I$ is never below the tangent line to the graph at $(c,f(c))$. I do not understand the question and I ...
1
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1answer
124 views

indicator function of an entire function of finite exponential type?

Let $\Phi(z)$ be an entire function of finite exponential type. The indicator function of $\Phi(z)$ is defined as $$ ...
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2answers
143 views

Is there a function than transitions smoothly between two values, over an interval and remains constant elsewhere?

Specifically, does it exists an infinitely differentiable function $f:R \to R$ that meets the following conditions?. $f(x)=0$ if $x \le 0$. $f(x)=1$ if $x \ge 1$. Physical interpretation: Is there ...
3
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4answers
49 views

Prove $f$ is discontinuous on $r_n$?

Let $r_n$ be a sequence of all of the rational numbers and $$f(x)=\sum_{n:r_n<x}\frac{1}{2^n}.$$ Prove that $f$ is continuous on the irrationals;$f$ is discontinuous on the rational; ...
4
votes
1answer
67 views

Is Riesz measure an extension of product measure?

Suppose $X$ and $Y$ are compact Hausdorff spaces and $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ are finite regular Borel measure spaces. (By regular I mean that every measurable set can be ...
2
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2answers
142 views

Integral test for convergence proof

Can someone help me understand this proof? I don't understand why $f(n+1) = \int_n^{n+1}{f(n+1)}$ Thank you so much and I am sorry I have nothing else to contribute as I'm fearing it is a ...
0
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1answer
24 views

Using the Newton-Quotient

I am trying to prove that the function $f(x) = |8x^3 − 1|$ isn't differentiable at $x=\frac{1}{2}$. Now I know that I need to use the newton quotient to prove that the left hand limit is not equal to ...
3
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2answers
380 views

Increasing concave function

Let $f:[0,1]\rightarrow\mathbb{R}$ be a concave function with $f(1)=\sup_{t\in[0,1]} f(t)$. Then $f$ is non-decreasing in $[0,1]$. Does someone know how to prove this?
0
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1answer
17 views

Proving lebesgue increasing convergence thm…

Im not curious about the proof itself. Just want to know why For arbitrary $c<\int f d\lambda $, if $I>c \rightarrow I \geq \int f d\lambda$ I have seen this kind of argument in some of proof ...
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1answer
56 views

Intuitive explanation of the potential function of a vector field

Suppose I have some vector field $$\vec{f}(x,y)=\begin{pmatrix}A(x,y)\\B(x,y)\end{pmatrix}$$ then the potential function (if the field is conservative) can be found by integrating $A$ with respect ...
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3answers
1k views

Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?

Is Lipschitz's condition necessary condition or sufficient condition for existence of unique solution of an Initial Value Problem ? I saw in a book that it is sufficient condition. But I want an ...
0
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2answers
30 views

Find points in which grad(f)(x,y) = 0

I need some help with the following task: Given is $f(x,y) = (4x^2+y^2) \cdot e^{-x^2-4y^2}$ I have to a) find points $(x_0, y_0)$ for which $\vec \nabla(f)(x_0,y_0) = 0$. b) calculate eigenvalues ...
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0answers
35 views

Examine a function regarding differentiability

I have to prove that $f: \mathbb R^3 \setminus\{0\} \to R : f(x) = \frac{e^{-\alpha \lVert x \rVert_2 }}{\lVert x \rVert_2 }$ is differentiable . So, what I have to prove is that $ \lim_{x \to ...
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0answers
14 views

Determine the largest set inside $\mathbb{R}^2$ such that $f(x,y)=inf ( x^2 , y^2 ) \in C^{1}$.

Problem: determine the largest set inside $\mathbb{R}^2$ such that $f(x,y)=\text{inf} ( x^2 , y^2 ) \in C^{1}$ i don't have idea for it.
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1answer
72 views

Regarding the basic topology Ex, 2.21 , 3rd edition of Principles of Mathematical Analysis by Rudin

I am firstly confused about the example 2.21 (c) when as a special case, the nonempty finite set has discrete point elements and hence about (d) too: ' If I've no limit point in my metric space ...
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3answers
151 views

No continuous injective map $f: \mathbb{S}^1 \to \mathbb{R}$ [duplicate]

A friend asked me if there could be a continuous injective map $$f: \mathbb{S}^1 \to \mathbb{R}.$$ My intuition tells me no. Endow $\mathbb{S}^1$ with a topology $\mathscr{T}$ and fix a pole $x ...
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2answers
40 views

Determine if the convergence of $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ is uniform.

$f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ converges pointwise in a set $E = [0, \infty)$ to $f(x) = x$. This problem reminds me a lot of how $\frac{x}{n}$ fails to converge uniformly to $f=0$ on ...
0
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2answers
401 views

If sequence $x_n$ converges to $x$, prove that $\sqrt x_n$ converges to $\sqrt x$ [duplicate]

If sequence $x_n$ converges to $x$, prove that $\sqrt x_n$ converges to c. I know I have to estimate $| \sqrt x_n - \sqrt x |$. But I cannot start. Thanks
4
votes
2answers
91 views

$\lim_{n\to ∞} \left[\frac{f\left( x +\frac1n\right)}{ f(x)}\right]^n$

Could anyone solve this problem for me? Let f be a positive differentiable function on the internal $\left[\,0,\infty\right)$. $$\lim_{n\to ∞} \left[\frac{f\left( x +\frac1n\right)}{ ...
1
vote
1answer
73 views

If $x_{n}$ and $x_{n}y_{n}$ are bounded, does it follow that $y_{n}$ is bounded? [closed]

If $x_{n}$ and $x_{n}y_{n}$ are bounded, does it follow that $y_{n}$ is bounded? Attempt Let |$x_{n}| \leq C$ and |$x_{n}y_{n}| \leq C'$, then |$x_{n}y_{n}|$ $\leq$ $ |y_{n}|$ $\leq C'/C$. If ...
0
votes
2answers
94 views

How to find $\lim_{n\to ∞} \Big[ \Big(1+ \frac1n\Big)^n - \Big(1+\frac1n\Big) \Big]^{-n }$?

Could someone give me a hint on how to calculate this limit? $$\lim_{n\to ∞} \Big[ \Big(1+ \frac1n\Big)^n - \Big(1+\frac1n\Big) \Big]^{-n }$$ I tried taking the logarithm, but after that what ...
2
votes
3answers
32 views

Determine if $f_n(x) = n^2x(1-x^2)^n$ converges uniformly on $E=[0,1]$.

Determine if $f_n(x) = n^2x(1-x^2)^n$ converges uniformly on $E=[0,1]$. We can easily find the pointwise limit to be $f(x) = 0$ for all $x \in E$. $M_n = \sup\limits_{x\in E} |f_n(x)-f(x)| = ...
4
votes
3answers
57 views

Function that satisfies $\int_{2^{-n}}^{2^{-(n+1)}} f(x) dx = \int_{2^{-(n+1)}}^{2^{-(n+2)}} f(x) dx$

I was wondering if anyone would be able to help me find a function that satisfies this condition: $$\int_{2^{-n}}^{2^{-(n+1)}} f(x) dx = \int_{2^{-(n+1)}}^{2^{-(n+2)}} f(x) dx$$ It needs to be able ...