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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. .

145439 questions
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Need some help on baby Rudin theorem 6.15

Following is theorem 6.15 of baby Rudin: If $a
Daniel
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What is the difference between open ball and neighborhood in real analysis?

I'm learning real analysis. Open ball: The collection of points $x \in X$ satisfying $|x - x_{0}| < r$ is called the open ball of radius $r$ centered at $x_{0}$ Neighborhood: A neighborhood of $x_{0} \in X$ is an open ball of radius r > 0 in…
Jill Clover
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Prove $\sup(f+g) \le \sup f + \sup g$

Suppose $D$ is a nonempty bounded subset of reals. Let $f:D \to \mathbb R$ and $g:D \to \mathbb R$. Define $(f+g)(x)=f(x)+g(x)$. Prove $\sup(f+g)(D) \le \sup f(D) + \sup g(D)$ (also prove that $\sup (f+g)$ exists). I understand why this is the…
fhyve
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Infinite intersection of open sets

I need to prove that the infinite intersection of open sets may [must] not be open. I can show through examples that this is true, but this is not sufficient for a proof. - Can somebody give a formal proof ? Thanks.
user249018
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Does having a minimum along all lines imply that the function has a minimum?

Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$, with $f \in C^k$, $k \geq 2$. Suppose that $f$ has a local minimum at the origin along all lines. That is, for all $(x, y) \in \mathbb{R}^2$, the function $g_{x, y}(t) = f(tx, ty)$ has a local minimum at…
user182973
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Discontinuities of the derivative of a differentiable function on closed interval

I have a question about the corollary to theorem 5.12 in Rudin's Principles of Mathematical Analysis (page 108): Suppose $f$ is a real differentiable function on $[a,b]$ and suppose $f'(a)< \lambda < f'(b)$ then there is a point $x \in (a,b)$ such…
Danny
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What's the fastest way to tell if a function is uniformly continuous or not?

I have my real analysis final tomorrow and there are multiple choice questions. I'm wondering about a fast way to tell if a function is uniformly continuous or not. I know and understand the definition of uniform continuity, and I understand its…
user114014
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Supremum of Infimum and Infimum of Supremum

Let $X$ and $Y$ be nonempty sets, and let $f:X\times Y \rightarrow \mathbb{R}$ be a bounded function. a) Prove that $$\sup_{y\in Y}\left(\inf_{x\in X}f(x,y)\right)\leq \inf_{x\in X}\left(\sup_{y\in Y}f(x,y)\right)$$ b) Give an example (with proof)…
JohanLiebert
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How to show $E:=\{x\in [0,1]: |f^{-1}(\{x\})|\,\mbox{ is even} \}$ is countable?

Hi everybody I have seen the following question which I could not solve it, so I thought I can share the question with you and ask for help. Question: Let $f:[0,1]\to [0,1]$ be a continuous function such that $f(0)=0$ and $f(1)=1$. Moreover assume…
m.b
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1 answer

Simpler proof of the Hardy-Littlewood-Sobolev inequality in the inhomogeneous case

The Hardy-Littlewood-Sobolev inequality is the statement that there is a $C>0$ such that $$ \tag{HLS} \lVert f\ast \lvert\cdot\rvert^{-\alpha}\rVert_p\le C\lVert f\rVert_q, $$ for all $f\in L^q(\mathbb R^d)$, where the convolution is defined…
Giuseppe Negro
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Can the product of a sequence of numbers between 0 and 1 converge to positive?

Let $x_n \in (0,1)$, is it possible that $\prod_{n=1}^\infty x_n >0$ ? I think it isn't, because such small numbers multiplied together will become smaller and smaller, but I am not sure if there is a positive lower bound for the product. Thanks!
Tim
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Archimedean property

I've been studying the axiomatic definition of the real numbers, and there's one thing I'm not entirely sure about. I think I've understood that the Archimedean axiom is added in order to discard ordered complete fields containing infinitesimals…
Abel
  • 1,792
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Supremum of the Difference of Two Functions

Given two real-valued functions $f$ and $g$, is it true that $\sup(f-g) \geq \sup(f) - \sup(g)$
Elements
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Advantage of Lebesgue sigma-algebra over Borel?

What it says on the tin. Using the Borel $\sigma$-algebra on the reals instead of the Lebesgue $\sigma$-algebra has the advantage that it allows a broader class of measures, many of which are quite natural: For example the "uniform" measure on the…
unknought
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How does one prove that a multivariate function is univariate?

The question resembles How does one prove that a multivariate function is constant? but appears to be more difficult. Suppose that $u\colon\mathbb R^2\to\mathbb R$ is a continuous function such that at every point $(x,y)\in\mathbb R^2$ at least one…
user31373
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