Continuous Extension Theorems in Real Analysis Stackexchange
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For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.
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prove that $\int_0^1 (x-f(x))dx \leq 1-4b(1-a)$
Source: 2013 UofT competition, problem 3. Let $f(x)$ be a convex increasing real-valued function defined on the closed interval $[0,1]$ for which $f(0)=0, f(1)=1$. Suppose $0<a<1 $ and $b =f(a)$...
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Convergence of $\sum x^n$ in $C([0,1])$
Consider the space of continuous functions $C([0,1])$. Does the series $\sum_{n=1}^\infty x^n$ converge in this space? The series $\sum_{n=1}^\infty x^n$ converges if the sequence $(s_n)_{n=1}^\infty$...
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Cartesian product of smooth maps is smooth
Suppose $f,g \in C^\infty(X)$. I would like to show that $f \times g \in C^\infty(X \times X)$. This seems obvious, but I'm not sure how to actually prove it. I was thinking of using projection maps $...
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Proving that a real function is bounded
Let $f$ be a real function and $\alpha>1$. If $f$ satisfies the following inequality \begin{equation} f(t)\leq c+\int_{0}^{t}f^{\alpha}(s)ds \end{equation} where $c>0$. How can I prove that $f$ ...
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1 vote
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Proof that $a_{n}:=\sup\{|\frac{nx}{1+n^2x^2}|:x\in \mathbb{R}|\}$ $(n\in \mathbb{N})$ does not converge to $0$
I am trying to prove that the sequence $a_{n}:=\sup\big\{|\frac{nx}{1+n^2x^2}|:x\in \mathbb{R}|\big\}$ $(n\in \mathbb{N})$ does not converge to $0$. Is the following correct? Consider arbitrary $n\in \...
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1 vote
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For convex $f$, how to construct smooth $f_\varepsilon$ with $f- \varepsilon \leq f_\varepsilon \leq f\;$?
Let $f : \mathbb R_+ \to \mathbb R$ be a convex function such that $f(x) \to -\infty$. For $\varepsilon > 0$, is there a (standard) way to construct a convex function $f_\varepsilon \in C^\infty$ (...
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Extending Vitali's Covering Lemma to Countable Collections
I am trying to show an extension of the Vitali's Covering Lemma to countable collections, i.e. let $E$ be a set of finite outer measure and $F$ a collection of closed, bounded intervals that cover $E$ ...
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The probability that there exists finite N such that N products of i.i.d. random variable is smaller than some threshold is 1
Let $X$ be a random variable which has a probability distribution $\mathcal{P}$ where the support of $\mathcal{P}$ is (0,1). Let $\{X_n\}_{n\ge0}$ be a sequence of random variable where $X_{n}\overset{...
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1 answer
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Subsequence : Error in instructor's correction?
http://faculty.usiouxfalls.edu/sjc/fall_06/mat320_soln/hmk10.pdf Show that if ($x_n$) is unbounded, then there exists a subsequence ($x_{n_k}$) such that lim $1/x_{n_k} = 0$. Proof. By the Monotone ...
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Can we swap integral and limit?
$$ D_{\alpha}(p \| q)=\frac{\int ( \alpha p(x)+(1-\alpha) q(x)-[p(x)]^{\alpha}[q(x)]^{1-\alpha}) d x}{\alpha(1-\alpha)},\;\alpha\in(0;1).$$ Here $p(x)$ and $q(x)$ are probability densities functions. ...
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Relationship between Big O vs little o
I'm having trouble with some statements using the Big $O$ and little $o$ notations. Let $f(x)=O(x^2), g(x)=O(x^3)$ for $x \rightarrow 0$. Which of the following statements are true: $f(x)=o(x)$ for $...
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2 votes
2 answers
33 views
$C_{1}f(x) \leq g(x) \leq C_{2}f(x)$ where $f,g $ be continuous and positive functions
Let $f$ and $g$ be continuous and positive functions on $[a,b]$. Then there are positive $C_{1},C_{2}$ such that $$ C_{1}f(x) \leq g(x) \leq C_{2}f(x). $$ I know that there are $\alpha_{1},\alpha_{2}...
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How do they get these two expressions (dθ/dx) and (dθ/dy)?
I tried deriving the tanθ with respective to x and y but couldn't arrive at the correct solutions. Could anyone please show or give a hint? Thank you
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Rudin's RCA Theorem $3.17$
There are the definitions which we need for the proof of the theorem: There is the theorem: If $X$ is a locally compact Hausdorff space, then $C_0(X)$ is the completion of $C_c(X)$, relative to the ...
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1 vote
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35 views
Confusion between $f(x)>f(0)$ or $f(x)<f(0)$ for a differentiable function.
If $f:(-1,1)\to \mathbb R$ is a differentiable function. Which of the following is necessarily true? If $f'(0)>0$, then $f(x)>f(0),\forall x\in(0,1)$. If $f'(0)<0$, then $f(x)<f(0),\...
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