A infimum of integral of a cubic function
Let $f(x)$ be a cubic function defined on $[0,1]$, and $\max_{x \in [0,1]}\left|f(x) \right|=1$. Find the infimum of $$C=\int_0^{1}\left|f(x) \right|\text{d}x.$$Note that $$C=\inf\int_0^{1}\left|f(x)...
View ArticleBorel sigma algebra of extended Real Line
I'm trying to prove the following (where $\overline{\mathbb{R}} = \mathbb{R} \cup \{+\infty, -\infty \}$): The Borel sigma algebra $\mathcal{B}(\overline{\mathbb{R}})$ is the sigma algebra generated by...
View ArticleTrue or False? Minimum of a convex function on a ball is the point on the...
I am wondering whether this is true or false, since I can't seem to find any counterexamples.Suppose $f:\mathbb{R}^d\to\mathbb{R}$ is convex and has a minimizer $\mathbf{x}^{\star}$. Let...
View ArticleProving the cut property
Question: If $A$and $B$are nonempty, disjoint sets with $A\cup B=\mathbb{R}$and $a<b$for all $a∈A$and $b∈B$. Show that there is a real number $c$ such that $x < c $ implies $x ∈ A$, and $x >...
View Articleif integral of a continuous function on a closed interval is zero, so is the...
let $f:[0,1]\to R$ be continuous, s.t $\int_a^b f(x)dx=0$ for $0 \le a \le b \le 1$. Show $f$ is $0$.this is the exact text of a qualifying exam question.I do not know if $f$ is bounded, so continuity...
View ArticleCompleteness of the sum of two $L^p $ spaces
Q:Suppose $L^{p_0}+L^{p_1}$ is defined as the vector space of measurable functions $f$ on a measure space $X$,that can be written as a sum $f=f_0+f_1$ with $f_0\in L^{p_0}$ and $f_1\in L^{p_1}$....
View Articlesequences based upon the cantor space
Considering the cantor space i.e. $2^\mathbb{N}$, what would be examples elements $x$ in the cantor space such that $\lim \frac{|\{m < n:x(m) = 1\}|}{n}$ does not converge. If $x$ has a finite...
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Existence of the supremum can't be proven using field and order axioms
I'm reading through "Introduction to Real Analysis" by Robert G. Bartle and Donald R. Sherbert, fourth edition, page 39.But how does he know that the existence of the supremum can't be proven using the...
View ArticleSame inner product with functions in $C^{\infty}_c(\Omega)$ implies same...
Let $\Omega\subset\mathbb{R}^n$ be an open set. It is true that if $u,v\in H^1(\Omega)$ and$$(u,\phi)_{H^1(\Omega)}=(v,\phi)_{H^1(\Omega)},\ \forall\phi\in C^{\infty}_c(\Omega)$$then $u=v$? Here I have...
View ArticlePositive and negative portions of conditionally convergent series
I want to prove that for a conditionally convergent series $\sum_{n = 1}^{\infty} a_n$, that its positive subseries (Let $K := \{k \vert a_k > 0\}$, and the positive subseries is $\sum_{j \in K}...
View ArticleThe distance between a closed set and a bounded closed set which are disjoint...
Prove/disprove the distance between a closed set and a bounded closed set which are disjoint has a positive lower bound (only consider the metric space $\mathbf{R}^n$)$A$ is a closed set in...
View Articleif $f,g$ are continuous and increasing, $\exists! (u,v) \in \mathbb{R}$ so...
I'm going in circles to show the unicity$h: u \mapsto u + f(u - v)$ is strictly increasing with v fixed and by IVT $\exists! u \in \mathbb{R}$ such that $h(u)=0$ (easy to show)Same is true with $d: u...
View ArticleAsymptotic behaviour of athe function $\displaystyle\exp\left( \beta...
I am struggling to get from the last step of the proof of Lemma 2.4 to equation 2.17 of this paper.The problem is the following:let $a>0$ and $\beta>0$ be two positive numbers and $I_\beta$ the...
View ArticleIf $\sum\limits_{n=0}^{\infty} a_n$ exists (finitely or infinitely) then,...
Fact:"If limit of a real sequence $\{a_n\}_{n \in \mathbb{N}}$ exists then the limit of all subsequences of $\{a_n\}_{n \in \mathbb{N}}$ exist and is equal to the limit of the sequence $\{a_n\}_{n \in...
View ArticleDoes a stretch of a set $E\subset\mathbb{R}$ about a condensation point of...
A point $p$ in a metric space $X$ is said to be a condensation point of a set $E\subset X$ if every neighbourhood of $p$ contains uncountably many points of $E.$ [Note that $p$ need not be in...
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Existence and continuity of one-side derivative inplies differentiable on the...
I'm dealing with a calculus exercise. Let me first explain my notation: I say $f_-^\prime(x)$ exists provided the limit $\lim_{t\to x^-}\frac{f(t)-f(x)}{t-x}$ exists. Also we define four Dini...
View ArticleWhen is the function $f$ differentiable over irrationals?
Consider the function$$f\left( x \right) =\begin{cases}0,x\in \mathbb{Q} ^c\\a_n,x=\frac{m}{n}\in \mathbb{Q}\\\end{cases}$$where $(m,n)=1$. My question is, when is $f$ differentiable over...
View ArticleA Nowhere Locally Bounded and Non-Riemann Integrable Function
Here is the Example 1.16 of Sheldon Axler's Measure, Integration & Real Analysis.Let $r_1,r_2,\dots$ be a sequence that includes each rational number in $(0,1)$ exactly once and that includes no...
View ArticleIs there any danger in supposing m>n in the Cauchy sequence conditions?
$$\forall \epsilon > 0,\ \text{let} \ N=\frac{1}{\sqrt{\epsilon}}.$$$$\text{So}, \ m>n>N \implies \frac{1}{m^2}<\epsilon. $$$$|s_n-s_m| =...
View ArticleHow to evaluate this integral? $\int^1_0\frac{(\frac{1}{2}-x)\ln...
$$\int^1_0\frac{\left(\frac{1}{2}-x\right)\ln (1-x)}{x^2-x+1}\mathrm{d}x$$The indefinite integral didn't seem to be helpful...which include the $\operatorname{Li}(x)$.Also, I can't set up a parameter...
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