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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3
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3answers
331 views

a linear differential equation with periodic coefficients

Let $$y' = a(x) y + b(x)$$ be a linear differential equation with continuous, periodic coefficients $a, b: \mathbb{R} \to \mathbb{R}$ that both have a period of $T > 0$. Also, we assume that ...
5
votes
5answers
81 views

Show $\sum n e^{-na}$ converges for $a>0$

Is there any test or property in particular I can use to show $ \sum n e^{-n a}$ is convergent for $a>0$ ? I think it is obvious that from looking at the function that this is convergent, since ...
0
votes
0answers
33 views

How to modify Tikhonov regularization?

Consider a linear map $f: X \rightarrow Y$ and let $F$ is a matrix of $f$ and $b$ is one element of $Y$. Our goal is to obtain the element of $X$ corresponding to $b$. In ideal case we can get the ...
2
votes
1answer
74 views

Showing Uniform convergence of $\frac{n x}{1 + n \sin(x)}$

I want to prove for all $a\in \left(0,\frac{\pi}{2}\right]$, $ \ f_n\to f$ uniformly on $\left[a,\frac{\pi}{2}\right]$. Also, how is this different from $f_n \to f$ uniformly on $\left(0, ...
0
votes
0answers
61 views

Proving that region bounded by y=0 and continuous function is a Jordan region.

How do I prove the following? Let $f(x)$ be continuous on $[a,b]$ and let $A=\{(x,y): x \in [a,b] \text{ and } 0 \le y \le f(x)\}$. Prove that $A$ is a Jordan region. I know that I can show ...
1
vote
1answer
56 views

Uniform Convergence of $(\frac{x}{1+x})^n$

I have an exercise based on the exercise 8 of page 41 in Complex Analysis of Ahlfors. In that exercise they ask for the values of x in which the following series converges: $$ ...
1
vote
2answers
92 views

Find all continuous real valued function such that $(f(x))^2+C=\int\limits_0^xf(t)dt$ [duplicate]

Find all continuous real valued function such that $$(f(x))^2+C=\int\limits_0^xf(t)dt$$ for some $C\in\mathbb{R}$ If I set $F(x)=\int\limits_0^xf(t)dt$ then $F$ is differentiable and ...
2
votes
1answer
48 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
2
votes
2answers
56 views

$\mathrm{d}f(x,t)$ this way $d\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} \,dt+\frac{\partial f}{\partial x}\,dx$?

If $dX_t=a_t \,dt$ the next procedure is correct? $$\mathrm{d}\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} dt+\frac{\partial f}{\partial x}dx=\frac{\partial f}{\partial t} dt+\frac{\partial ...
2
votes
0answers
114 views

Finding the maximum of two functions with complicated formulas

Let $$ f(\omega)=1+\frac{m(a+\omega^2)}{a^2+\omega^2}+\alpha\left(\frac{a^2+\omega^2-ma}{a^2+\omega^2}\right)\cos(\omega\tau)+\frac{\alpha m\omega}{a^2+\omega^2}\sin(\omega\tau)\;, $$ and $$ ...
2
votes
3answers
68 views

Polynomial of degree $2$ has at most $2$ roots

Suppose that $P: \mathbb{R} \rightarrow \mathbb{R}$ is a real polynomial of degree exactly $2$. Prove $P$ has at most two roots. Let $P(x)=a_2 x^2 +a_1 x +a_0$ for all $x \in \mathbb{R}$. I tried to ...
0
votes
1answer
15 views

How I find a suitable increment value

If I have three variables $x,y$ and $z$, where $x\lt z$ and $y\lt z$, then I need to make each value of $x$ and $y$ equal to or approximately equal to $z$ by adding a ratio of another variable, for ...
3
votes
2answers
129 views

Does $\frac{nx}{1+n \sin(x)}$ converge uniformly on $[a,\pi/2]$ for all $a \in (0,\pi/2]$?

Edit: the question had some missing details. It should read as follows: Prove for all $a \in (0,\frac{\pi}{2}]$, $f_n \rightarrow f$ uniformly on $[a,\frac{\pi}{2}]$. Here $$f_n(x) = \frac{n x}{1 ...
0
votes
1answer
42 views

Bounded sequence of positive numbers

Suppose that $\{x_n\}$ is a sequence of positive real numbers so that $\lim \frac{x_{n+1}}{x_n}=L<1$. Then show that for $n$ large enough and for some $C>0$ we have $0<x_n<Cr^n$. I have ...
0
votes
2answers
55 views

How prove this inequaliy $\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt $

let $f \in C^1([a, b])$ with $a, b \in \mathbb{R}, a < b$ show that $$\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt$$ I've tried to use the ...
0
votes
1answer
59 views

Real Analysis Question- Differentiability on an interval

So this is the question I am trying to answer... At $(-2,0)$ and $(0,2)$ we just differentiate using normal rules of calculus, yes? Here is my attempt for at $x=0$. Is this correct? For d)I think ...
2
votes
2answers
66 views

For which $x\in\mathbb{R}$ is the series of general term $a_n = x^{n!}$ convergent?

I firstly found the simplified form of $\frac{a_{n+1}}{a_n} = |x|\cdot|x^n|$ and used this to establish the end points $-1\lt x\lt 1$. I then tested the end points by finding the limit to infinity of ...
0
votes
1answer
47 views

Lef $f'$ be integrable and $f(0)=0$. Show that $|f(x)|\leq\sqrt{\int\limits_0^1|f'(t)|^2dt}$

Lef $f$ be a fuction such that $f'$ is integrable in $[0,1]$, $f(0)=0$. Show that $$|f(x)|\leq\sqrt{\int\limits_0^1|f'(t)|^2dt}$$ forall $x\in[0,1]$ I did $f(x)^2\leq\int\limits_0^1(f'(t))^2dt$ ...
-2
votes
2answers
44 views

Differential equation maximal interval and solution [closed]

Consider the differential equation $y' = 1 - y^2$. First, is $y(x) = 1$ the only constant solution? I now want to solve the equation for the initial value problem $y(0) = y_0$, with $y_0 > 1$. ...
3
votes
1answer
55 views

Inverse Function Theorem for Banach Spaces

In the middle of a proof of the Inverse Function Theorem (namely, the proof of Baby Rudin), we use the fact that if $A$ is invertible and: $$ ||B-A||~||A^{-1}|| <1$$ then $B$ is invertible. The ...
2
votes
3answers
71 views

Real Analysis - differentiable

$f:[0,\infty]\rightarrow \mathbb{R}$ is twice differentiable. If $f''$ is bounded and exists the limit of $f(x)$ at infinity, then $\lim_{x\rightarrow \infty}f'(x)=0$. I tried to use the Taylor's ...
0
votes
1answer
44 views

A domain in $\mathbb{R}^n$ with $C^2$-boundary satisfies an “outer spherical condition”

Let $\Omega\subseteq\mathbb{R}^n$ be a domain and $\partial\Omega\in C^2$, i.e. $\Omega=\overline{\Omega}^\circ$ For all $x_0\in\partial\Omega$, there exists a neighbourhood $U\subseteq\mathbb{R}^n$ ...
2
votes
1answer
79 views

Let $f,g$ be differentiable functions such that $\int\limits_0^{f(x)}f(t)g(t)dt=g(f(x))$. Show that $g(0)=0$

Let $f,g$ be differentiable functions such that $$\int\limits_0^{f(x)}f(t)g(t)dt=g(f(x))$$ Show that $g(0)=0$ I know it is done if $f(x)=0$ for some $x$, but I just have ...
0
votes
0answers
36 views

What is the norm for the product of normed spaces?

Suppose $(\Omega, \Sigma ,\lambda)$ is a probability space and $X_i=L^2(\Omega,\Sigma,\lambda ,[0,1])$ with norm $||.||_{L^2}$ for all $i\in I$, $I$ is finite. Is there any natural norm for the ...
5
votes
0answers
52 views

Importance of compactness in Rudin problem.

Okay the problem goes like this: Suppose X, Y, Z are metric spaces, and Y is compact. Let $f$ map X into Y, let g be a continuous one-to-one mapping of Y into Z, and put $h(x)=g(f(x))$ for $x\in ...
1
vote
1answer
44 views

L^p spaces are separable and complete but not compact?

Where is the mistake in my reasoning?: Let X be a separable metric space, then for every $p\in [1,\infty)$ and for every borel measure $\mu$ on $X$: $L^p_{\mu}(X)$ is separable. Therefore by a ...
-1
votes
1answer
45 views

Why is there a subsequence of $(x_n)$ that converges to some point $y$ in $\mathbb R^p$?

A subset $A\subseteq\mathbb R^p$ is compact iff for every sequence $(x_n)$ in $A$ there is a subsequence $(x_{n_k})$ which converges to a point of $A$. I understand the whole proof of the above ...
1
vote
2answers
65 views

Show every chain has an upperbound?

Sometimes I feel like proofs like this are pointless. I mean, if we have a partially ordered subset, it seems automatically true that you have a max element. 1) Either you have an infinite sequence ...
2
votes
1answer
58 views

Example for the benefit from monotone convergence

I want to see a (preferably simple) example where I can apply monotone convergence to a sequence of functions $f_n$ but where I cant exchange limitation and integration in terms of the Riemann ...
0
votes
3answers
124 views

Open sets and compact spaces

I am reading through Rudin's Principles of Mathematical Analysis and had a few related questions. First, Rudin defines an open set, $E$, as a set such that every point is an interior point. A point ...
1
vote
1answer
32 views

does constant convexity assures global minimum

I have the following question: Consider a function $f:R^n \longrightarrow R$, s.t.: there is a point $x_0 \in R^n$ s.t. $\frac{\partial f}{\partial x^k} =0$ $\forall k$. the hessian matrix ...
11
votes
1answer
86 views

Diffeomorphism-invariant spaces of smooth functions

Let's start with an interesting story. In his celebrated Partial Differential Relations (p. 146), the great Misha Gromov gives a nice exercise of which the following is a (strict) part. Exercise. ...
2
votes
1answer
90 views

Find a cut-off function in a ball.

Let $0\le r< R\le 1$. How do we find a function $\eta\in C^1(\mathbb{R})$ such that $\eta=1$ in $B_r$ (the ball center at $0$ and radius $r$) and $\eta=0$ outside $B_R$ and $|D\eta|\le ...
3
votes
3answers
42 views

Boundedness theorem question proof check

Here is an attempt at a solution: Since $f(x)>0$, $f(x)>\delta$ for all x between $1$ and $2$ Is this correct?
1
vote
2answers
62 views

Show that $A$ is open in $\mathbb R$

I got this question in a test earlier today. I know it is a very small question, since it only counted 2 marks, but for some reason I simply could not get it?? Let $f:\mathbb R \to \mathbb R$ be ...
1
vote
3answers
72 views

How many continuous involutions on $\mathbb R$ are there? [duplicate]

An involution is a function that satisfies the following: $f = f^{-1}$ MY question is how many involutions can you find in the set of real functions, and how would you go about solving that problem? ...
2
votes
0answers
48 views

Continuous function on compact subset of $\mathbb R$ to itself has a fixed point.

Let $f:[a,b] \to [a,b]$ be continuous. Then $f$ has at least a fixed point. I read the following proof from Limaye book. Define $F(x)=f(x)-x.$ Since $a \leq f(x) \leq b,\ \quad F(a)\leq 0 \ \quad ...
-3
votes
1answer
36 views

how to construct a monotonic function on a closed interval which is discontinuous at each end points [closed]

How to construct a monotonic function on a [0,1] which is discontinuous at each end points?
1
vote
1answer
63 views

modulation-translation operator continuous in $L^{p}$ norm?

We put, $T_{y}f(x):=f(x-y), \ (x, y\in \mathbb R^{n}).$ It is well-known that $\|T_yf-f\|_{L^{p}} \to 0$ as $y\to 0$ for $1\leq p <\infty.$ Next we put, $M_tT_yf(x):= f(x-ty) e^{i t (x\cdot y)}, ...
1
vote
1answer
39 views

Prove or disprove regarding continuity of $f$ and $g$

Prove or disprove: Let, $f,g:[a,b]\to \mathbb R$ be continuous in $[a,b]$ and are non-zero at any point. There exists $c\in [a,b]$ such that $$g(c)\int_a^bf(x)\,dx=f(c)\int_a^b g(x)\,dx.$$ ...
-1
votes
1answer
49 views

Find the point-wise limit of this sequence of function $\{f_n(x)\}$.

Consider the sequence of function $\{f_n(x)\}$ in $[0,1]$ where , $$f_n(x)=\begin{cases}0 & \text{ if } x=0\\n^2x & \text{ if } x\in [0,\frac{1}{n}]\\-n^2x+n^2 & \text{ if } x\in ...
2
votes
2answers
106 views

Why Cantor set removes one third?

I found the derivation of Cantor-like set in Understanding Analysis by Abbott. There he removes one fourth, and most properties (length, cardinality, compactness, uncountableness) are preserved ...
1
vote
0answers
23 views

convex function with Hessian measure $D^2 f \leqslant \lambda$ $\lambda$-concave?

Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
2
votes
1answer
251 views

Proof Norm is Continuous

Someone just asked me why the norm of a normed space is continuous, and the answer I gave them satisfied them, but I'm not sure if it should. Something seems amiss. Let $\rho: X \to \mathbb{R}^+_0$ ...
1
vote
1answer
56 views

Can a real value function, defined for every real number, have finite (or numerable) points of continuity?

Can a real value function, defined for every real number, have finite (or countable) points of continuity ? As for the not countable case, the answer is trivial: any polynomial has not countable ...
1
vote
1answer
94 views

Find all continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $(f(x))^2+8=\int\limits_0^xf(t)dt$

Find all continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$(f(x))^2+8=\int\limits_0^xf(t)dt$$ If I set $F(x)=\int\limits_0^xf(t)dt$ then $F$ is differentiable and $F'(x)=f(x)$, ...
2
votes
1answer
34 views

How can I find these partial derivatives?

I'm reading a book which gives this function $f(x,y)=x^2y/(x^2+y^2)$ if $(x,y)\neq (0,0)$ and $f(0,0)=0$ as a $C^1$ function in $\mathbb R^2-\{(0,0)\}$, continuous in $(0,0)$ and it has the partial ...
0
votes
1answer
735 views

Differences between real and complex analysis?

To start with, real analysis deals with numbers along the (one dimensional) number line, while complex analysis deals with numbers along two dimensions, real and imaginary, Cartesian style. Could this ...
1
vote
2answers
56 views

If $\frac{\partial F^i}{\partial x^j}=0$ on a connected open set, is $F$ constant?

Let $U$ be open in $\mathbb{R}^n$ and let $$F:U\to \mathbb{R}^m$$ be a smooth map, i.e. $F\in C^\infty(U)$. It is easy to prove that if $U$ is convex and $$\frac{\partial F^i}{\partial x^j}=0\tag{1}$$ ...
0
votes
0answers
56 views

Is the linear operators must be invertible to from a category?

I am trying to understand the concept of category in mathematics. For example the following link talks about category $Lin$ which is an Abelian category. ...