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Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

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How to show that a polynomial has real root between two given values?

Let $C_0,C_1,\ldots,C_n$ are real constants. It is given that $$C_0 + \frac{C_1}{2} + \ldots + \frac{C_{n-1}}{n} + \frac{C_n}{n+1}= 0$$ We need to prove that the equation $C_0 + C_1 x + \ldots + C_{n-1} x^{n-1} + C_{n}x^{n} = 0$ has at least one…
Pradipta
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Lipschitz continuity of bilinear function

For a bilinear function $T$, it can be shown that $\lVert T(x,y)\rVert\leq C \lVert x\rVert \lVert y \rVert$ I saw some books say a bilinear function T is Lipschitz with Lipschitz constant $C$ given the above inequality holds. Now I'm confused…
user11869
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Sequence of Polynomials and Weierstrass's Approximation Theorem

I've been stuck on the following problem for a some time: Let $f$ be a continuous function on $[a,b]$. Show that there exists a sequence $(p_n)$ of polynomials such that $p_n \to f$ uniformly on $[a,b]$ and such that $p_n(a) = f(a)$ for all $n$.…
Student
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Let $f:[0,1]\to[0,1]$ be continuous then $f$ assumes the value $\int_0^1 f^2(t)dt$ somewhere in $[0, 1].$

True/False test: Let $f:[0,1]\to[0,1]$ be continuous then $f$ assumes the value $\int_0^1 f^2(t)dt$ somewhere in $[0, 1].$ $$f:[0,1]\to[0,1]\implies f^2:[0,1]\to[0,1]\implies 0\le\int_0^1 f^2(t)dt\le1$$ So it's true. but the paper says the statement…
Sriti Mallick
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Definition of bounded Set

I read in one book: The following statements about an upper bound $u$ of a set $S$ are equivalent: if $v$ is any upper bound of $S$, then u $\leq$ v. if $z < u$, then $z$ is not an upper bound of $S$. if $z < u$, then $\exists \ s_z \in S$ such…
mathguruu
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The zero of a continuously differentiable function is zero-measured

I'm tring to prove the following statement: Suppose $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is continuously differentiable, and for any $(x_0,y_0)\in \mathbb{R}^2$, we have $$ \frac{\partial f}{\partial x}(x_0,y_0)+\frac{\partial f}{\partial…
Robin
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Showing a sequence of functions converges to a Gaussian

Suppose we have a thrice continuously differentiable function such that the following hold: $f(0)=1$, $f^\prime(0)=0$, and $f^{\prime\prime}(0)=-1$. There exists $\alpha>0$ such that $\vert f(x)\vert\leq \exp(-\alpha x^2)$ Then,…
icurays1
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Find the minimal surface, integral over which for some function gives a certain value

So here is the problem: Let's say we have some function $f(x,y)$ And we have some value $a$, where $a =\int_S f(x,y)dxdy$ From the set of surfaces $\{S\}$ where the above is true, find the minimal $S'$. I realise that this might possibly be…
Jonas Petraitis
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Young's and Peter-Paul's inequalities

Following the idea from Jarchow 1981, pp. 47–55, let's retell the whole story in case some wishes to be complete. Let $1
MathArt
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Real Analysis - A sequence that has no convergent subsequence

How to prove that the sequence $\{a_n\}$ has no convergent subsequence if and only if $|a_n|$ approaches infinity? The forward direction is obvious, how to prove the other direction?
ysen07
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the continuity of the primitive

Good day! Need to prove the existence of $t\in [0,1]$ such that $$\int\limits_{0}^{t} f(x)\,dx = \frac{1}{2} \int\limits_{0}^{1} f(x)\,dx,$$ where $f$ is integrable. My solution: $$F(t)=\int\limits_{0}^{t}…
Vasili_Petrov
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Question about a proof of Bolzano-Weierstrass theorem

Let $(a_n)$ be bounded and $b > 0$ s.t. for all $n \in \mathbb N, \ |a_n| < b.$ Then $b \in \mathbb R$ with $\{n \in \mathbb N: a_n \le b\} = \mathbb N$ and $\{n \in \mathbb N: a_n \le -b\} = \varnothing.$ Thus $b \in X = \{x \in \mathbb R: \{n \in…
user1008945
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Proving $\left|\frac{r-x}{1-xr}\right| \leq r$ for $0\leq x\leq r$.

For $0
Jean
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Showing $F_n$ uniformly converges to $F$ if $f_n\to f$ and $F_n$ is the integral of $f_n$

Showing $F_n$ uniformly converges to $F$ if $f_n\to f$ and $F_n$ is the integral of $f_n$. Is my thought process okay? This is the problem: Does it suffice to use the definition of uniform convergence, and that $\sup(F_n-F) = $ integral($f_n-f$) =…
factos7
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Show that $x\mapsto \frac{\sin(x)-\sin(y)}{x-y}$ is increasing

Let $x,y\in [-\pi,0]$. How to show that $$ x\mapsto \frac{\sin(x)-\sin(y)}{x-y} $$ is increasing? After differentiating it, I get another problem to prove: $$ \frac{\sin(x)-\sin(y)}{x-y}\leq \cos(x), $$ in which I do not know how to approach it. I…
UnknownW
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