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
318 views

Show that $(0,1)$ is open in $\mathbb{R}$ and that $[0,1]$ is not open in $\mathbb{R}$

My Question: I want to show that the interval $(0,1)$ is open in $\mathbb{R}$ and that $[0,1]$ is not open in $\mathbb{R}$. I proved the latter, but I felt that it was clumsy and could be refined. If ...
3
votes
1answer
44 views

Derivative of function between sets of matrices

Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Let $\phi:M_{k,n}\rightarrow S_k$ be the map ...
7
votes
2answers
153 views

Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$

Let ${\rm f}:\left[a, b\right]\to\mathbb{R}$ be twice differentiable, and suppose $$\lim_{x\to a^{+}} \frac{{\rm f}\left(x\right) - {\rm f}\left(a\right)}{x - a} = \lim_{x\to b^{-}}\frac{{\rm ...
8
votes
2answers
157 views

Is it possible that all subseries converge to irrationals?

Does there exists a positive decreasing sequence $\{a_i\}$ with $\sum_{i\in\mathbb{N}} a_i$ convergent, such that $\forall I\subset\mathbb{N},\sum_{i\in I}a_i$ is an irrational number? Such an ...
2
votes
0answers
52 views

Measurability of functions and Daniell integral

I have just completed a few sessions of self-study of Daniell integral and it naturally extends Riemann integral and I can also see how to use it to "create" measures. However, I am starting to get ...
8
votes
1answer
127 views

Continuous bijections between $\mathbb{R}^2$ and $\mathbb{R}^2 \setminus \{(0,0)\}$

It is well-known that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus \{(0,0)\}$. I have two questions. a) Does there exist a continuous bijection $f: \mathbb{R}^2 \to \mathbb{R}^2 ...
3
votes
3answers
204 views

Continuous map from $[0,1]$ to $[0,1].$

Let $f$ be a continuous map from $[0,1]$ to $[0,1].$ Show that there exists $x$ with $f(x)=x. $ I have $f$ being a continuous map from $[0,1]$ to $[0,1]$ thus $f: [0,1]\to [0,1]$. Then I know from ...
0
votes
2answers
96 views

compact set boundary condition prove

the question is: Prove that a set E ⊂ R is open if and only if E is not contains any of its boundary points. I'm kind of not sure with good enough(?) proof... so please look at my several ...
3
votes
2answers
1k views

Closed ball is not compact

Show that the closed ball in $C([0,1])$ of center $0$ and radius $1$ is not compact. I thought it will be compact since every closed and bounded set in $\mathbb{R}$ is compact? Why is it not compact ...
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0answers
81 views

Closed Subset of $C[a,b]$

Show that the following subset $D$ of $C[a,b]$ under the $max$ norm is a closed subset where every function $f \in D$ is such that $\int\limits_a^b f'(x)^2 + f(x)^2 dx \leq k$ Attempt at a solution: ...
1
vote
2answers
93 views

Will somebody look over my real-analysis/calculus solution

Evaluate $\lim_{x\to 0}\frac{1}{x} \int_0^{x} \sqrt{9+t^2}\mathrm{d}t$. Immediately, taking, $$\begin{align} &\lim_{x\to 0}\frac{1}{x} \int_0^{x} \sqrt{9+t^2}\mathrm{d}t \\ ...
2
votes
2answers
99 views

If $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, 1-1, and on-to, then f maps Borel Sets to Borel Sets [duplicate]

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous, on-to, and 1-1. Prove that if $A$ is a Borel set, then $f(A)$ is a Borel set.
4
votes
1answer
73 views

Non isolated minimum

Consider the $C^2$ function $F:\mathbb{R}^k \rightarrow \mathbb{R}^k$ is it possible for it to be such that $x_n$ is a strict local minimizer for all $n$ and $x_n \rightarrow x$ where $x$ is a strict ...
0
votes
1answer
47 views

Inverse image of codomain

Is it true that if $f: X \rightarrow Y$ is continuous function then $f^{-1}[Y] = X$ ? I suppose that it is true and I try prove it: Suppose that $f^{-1}[Y] \neq X$ so exsists $x$ such that $ x \in ...
0
votes
1answer
67 views

almost everywhere convergence

Let $E\subseteq\mathbb{R}^l$ be s.t. $E$ is Lebesgue measurable and $m(E)>0$. Let for all $k\in\mathbb{N}$, $f_k:E\rightarrow \mathbb{R}$ be measurable functions. If for all $\epsilon>0$ we can ...
2
votes
3answers
124 views

How to prove a function has no maximum

I have a function: $$p\cdot(w+6s)^2+(1-p)\cdot(w-s)^2$$ where $p\in(0,1)$, $w>0$ and $s\geq0$ is a choice variable. I am looking for the maximum of the function with respect to $s$, but can ...
4
votes
2answers
114 views

“Rising sun” function

Let $f:[0,1] \to \mathbb R$ be bounded and $$ f_\odot :[0,1] \to \mathbb R: x \mapsto \sup \{f(y) : y \in [x,1] \} $$ This is well defined since $f$ is bouned. Claim: If $f$ is continuous then ...
0
votes
1answer
79 views

some weak conditions about Vitali convergence theorem

As we have known,the Vitali convergence theorem is stated: Let $(X,\mathbb{M},\mu)$ be a positive measure space.If (i)$\mu(X)<\infty$; (ii)$\{f_n\}$ is uniformly integrable; (iii)$f_n(x)\to ...
1
vote
1answer
74 views

Prove that d(.;A) is continuous

I need help with the following proof, which my professor added for practice (but not as homework). I am completely lost here. Let $A$ be a nonempty subset of a metric space $X$. Define $d(\cdot ,A) : ...
1
vote
1answer
31 views

Lebesgue measurability of a set considering its boundary

For any set $E\subset R^2$,the boundary $\partial E$ of $E$ is the closure of $E$ minus the interior of $E$. Then $E$ is Lebesgue measurable whenever $m(\partial E)=0$. How can I start with it?
1
vote
1answer
94 views

Subsets of real line without accumulation points; also, accumulation points of irrationals

I had to answer three questions about accumulation points. I think my work is correct, but I'd appreciate if someone would look over them for me. (I'm not sure if I read question 2 correctly.) In my ...
8
votes
1answer
212 views

Question on Riemann sums

Question is : What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$ So, at first sight i am afraid this is not even related to Riemann integration of ...
3
votes
1answer
113 views

Uniform convergence Check

Question is to check : I guess i am through with $(c)$ i.e., i see it as an infinite geometric series and say that : $$x^2+\frac{x^2}{1+x^2}+\frac{x^2}{(1+x^2)^2}+\dots = ...
0
votes
3answers
122 views

Does (Riemann) integrability of a function on an interval imply its integrability on every subinterval?

For example, if $f$ is integrable on $[0,3]$, is it also integrable on $[1,2]$? I tried thinking of a counterexample but couldn't, since I've only learned what implies integrability but not what ...
1
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1answer
48 views

When is something coordinate independent

One defines a critical point of a function $f$ to be a point where all its partial derivatives vanish. The critical points fall into two classes, degenerate and non-degenerate ones. A critical point ...
0
votes
1answer
25 views

Asymototics of a real sequence in a Riemann sum

Let $t<0$ and $f(k)\in O(|k|^{t})$ a real function, $k\in\mathbb{Z}$. We consider $$a_n\cdot \sum_{k=1}^n \frac{1}{n} \frac{f(k)}{n^t}$$ where $a_n\subset \mathbb{R}$ and ...
0
votes
1answer
62 views

Solving ODE by contraction mapping

Let a and c be real numbers. Solve the initial value problem y'(x) = ay(x), y(0) = c on the interval [0, 1/2a] with the help of the contraction mapping theorem. I understand that solving this ODE is ...
1
vote
1answer
344 views

The limit of sequence $(\frac{n \sin(2n)}{n^2 + \cos(n) + 4})$

I have trouble evaluating the limit of the sequence $(\frac{n \sin(2n)}{n^2 + \cos(n) + 4})$. Could anyone help me? Thank you!
2
votes
1answer
251 views

a function is continuous iff the graph is pathwise connected

Let f : [a, b] → R. By the graph of f we mean the set F = {(x, f(x)) : a ≤ x ≤ b} ⊆ R Prove that f is continuous if and only if the graph of f is a pathwise connected subset of the plane. I ...
2
votes
2answers
137 views

Zero, the Additive Identity, as the Multiplicative Annihilator

In the structures I have encountered so far, I have always seen a zero, which is usually defined as the additive identity. For example: $\exists 0 \in \mathbb{Z}$ s.t. $\forall a \in \mathbb{Z}, a ...
1
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2answers
102 views

Metric space question

Let (M,p) be a metric space and suppose that ${x_n}$ is a sequence in (M,p) so that $x_n -> x$ and $x_n->y$. prove x=y Let $E>0$. then, $p(x_n,x)->0$ $lim$ $n->inf$ ...
-3
votes
1answer
69 views

Proving various properties of metric spaces

Suppose that $p_1$ and $p_2$ are metrics on $M$. Prove that the following are also metrics: (a) $p = p_1 + p_2$ define $p_1(x,y) = |x-y|$ define $p_2(a,b) = |a-b|. p = p_1+p_2 = |x-y|+|a-b|$. But I ...
2
votes
1answer
168 views

$L^p$ Spaces, Young's Theorem, Convolutions, and Minkowski's Inequality

I need to show \begin{align} \|f*g\|_p \le \|f\|_p\|g\|_1 \end{align} By using the generalized Minkowski inequality instead of just Young's Theorem. I have spent a lot of time, but I keep hitting a ...
-1
votes
2answers
71 views

If $f$ is continuous, so is $f(x^3)$

Suppose $f:R\rightarrow R$ is differentiable and define $g(x)=x^2 f(x^3)$. Show $g$ is differentiable and compute $g'$. So I know how to do the proof, I just want to know that even though $f$ is ...
2
votes
1answer
109 views

The measure of the diagonal of a unit square in an alternative measure.

Usually, we say that the measure of the diagonal of a unit square is 0, but that's with the preassumption the measure is Lebesgue measure in $\mathbb{R}^2$. But what if we are talking about a strange ...
0
votes
2answers
76 views

proof of differentiatiable function

prove that x^(1/3) is differentiable at a with f(a)'=((a^(1/3))^-2)/3 for all a not equal to 0. I tried a epsilon-delta proof with limes theorem, and or that does not work or I am making somewhere ...
5
votes
1answer
173 views

looking for a diffeomorphism (not C1)

Let $f\colon\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ diffeomorphism with $f(B[0,1])\subset B[0,1]$ and $| \det f^{\prime}(x) |<1/2$ for all $x\in B[0,1]$ then for every continuous function ...
1
vote
1answer
34 views

Convolution composed with an invertible matrix

Let $T$ be an invertible $n \times n$ matrix and let $(h \circ T)(x)$ mean $h(Tx)$. Take functions $f,g$. Does it hold that $(f*g) \circ T = |det(T)| (f \circ T) * (g\circ T)?$ I have had some ...
2
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0answers
50 views

How can I check it?

$$f(x)=\sum_{n=1}^{\infty}\frac{\sin(2^{n}\pi x)}{n\cdot2^{n}}, $$ Is $f$ an absolutely continuous function? If yes how can I show it? If not how about on total variation of $f$ ?
2
votes
2answers
125 views

J-measurable sets and functions of class $C^1$

If $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is $C^1$ class and $det f^{\prime}(0)=0$ show that, when $r\rightarrow 0$ $$\dfrac{Vol(f(B[0,r]))}{ Vol(B[0,r])} \rightarrow 0$$ where $Vol(X)$ is ...
1
vote
1answer
32 views

If $f^{-1}(I)$ is connected for connected $I$ then $f$ is monotone.

Let $f:[0,1] \to \mathbb R$. Claim: If $f^{-1}(I)$ is connected for all connected $I \subseteq \mathbb R$ then $f$ is monotone. Assume $f(0) \leq f(1)$. I want to show that $f$ is monotone ...
7
votes
2answers
171 views

If $x\notin\mathbb Q$, then $\left|x-\frac{p}{q}\right|<\frac{1}{q^2}$ for infinitely many $\frac{p}{q}$?

This appears on problem 1 of chapter 1 in Stein & Shakarchi's Real Analysis: Given an irrational $x$, one can show (using the pigeon-hole principle, for example) that there are infinitely many ...
2
votes
3answers
203 views

angle between polynomials

let $v$ be the space of polynomials less than or equal to three and let $$\langle p,q\rangle = p(0)q(0)+p'(0)q'(0)+p(1)q(1)+p'(1)q'(1)$$ What is the angle between the polynomials $2x^3-3x^2$ ...
1
vote
1answer
170 views

Using Bernoulli's Inequality to prove an inequality

Using Bernoulli's Inequality show that $$\left(1+\frac{1}{k+1}\right)\left(1-\frac{1}{(k+1)^2}\right)^k\geq 1$$for all $k\in\mathbb{Z^+}$. My initial thought was to start be noting that ...
1
vote
0answers
31 views

proof that $\lim_{t \to x} \frac{1}{f(t)}=1/L?$ Given that $L \neq 0$

If $\lim_{t \to x} f(t)=L$ proof that $\lim_{t \to x} \frac{1}{f(t)}=1/L?$ Given that $L \neq 0$ My attempt for the case where $L>0$: $\large ...
0
votes
1answer
106 views

Which of these norms are equivalent to the canonical one

Regarding the space of continuously differentiable functions $C^1([0,1])$, I am wondering which of these norms are equivalent to the norm $||x||= ||x||_{\infty} + ||x'||_{\infty}$. The candidates are ...
2
votes
1answer
103 views

How to prove Riemann integrable with partitions

I'm quite stuck on how to prove that this function: $$ f(x) = \begin{cases} 1 & x \in [0,{1\over 2}) \\ x - {1\over 2} & x \in [{1 \over 2}, 1] \end{cases} $$ is Riemann integrable. I've tried ...
1
vote
1answer
407 views

Must a uniformly continuous function from $\mathbb{R}$ to $\mathbb{R}$ have a finite modulus of continuity?

I searched for similar questions and I could find a duplicate for this. NOTE: I am using the definition of "modulus of continuity" as found in Wikipedia for example ( ...
0
votes
0answers
34 views

Squares of differentiable functions

Suppose I have a function $f$ such that $f(x) \geq 0$ for all $x$. Suppose that $f^2$ is differentiable. Is it necessarily true that $f$ is differentiable? I was thinking of $$ f(x) = (-1)^x $$ but ...
0
votes
3answers
316 views

Rolle's Theorem Contradiction

Rolle's Theorem $x^3 - 3x +b$ Use Rolle's Theorem to prove that the equation $x^3 - 3x + b = 0$ has at most one root in the interval $[-1,1]$. Rolle's Theorem : Suppose f is a continuous real-valued ...