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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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Whether 'min' and 'division' are metric

Let $(X,d_1)$ and ($X,d_2)$ be metric spaces. Whether the following are again metrics on $X$ ? a) $d(x,y)=\text{min}\;\{d_1(x,y),d_2(x,y)\}$ b) $h(x,y)=\Big(\frac{d_1}{d_2}\Big)(x,y)$ where $x \neq y$ and $h(x,x)=0$ Actually the answer for the…
user444830
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Proving Rudin 1.6d (Exponentiation rule for real numbers)

I have been working through some of the early problems in Baby Rudin to prepare for a class next year, but am stuck on part (d) of question 1.6. Fix $b > 1$. (a). If $m, n, p, q$ are integers, $n > 0, q > 0,$ and $r = \frac{m}{n} = \frac{p}{q}$,…
SescoMath
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The convolution of two $L^2(\mathbb R)$ functions is continuous

Take $f$ and $g$ $\in L^2(\mathbb R)$, then I want to show that $\lim_{h \to 0} \int f(x-y-h)g(y)dy = f \ast g(x)$. My idea is to first take $f$ to be a continuous function with a compact support, then $f$ has to be uniformly continuous. Then we…
Adam
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Suppose $b \in \mathbb{R}$ and $|b| < \frac{1}{n}$ for every positive integer n. Prove that $b = 0$.

This comes from an exercise in Appendix C from Axler's Measure, Integration & Real Analysis. The following is my approach. Suppose $b \neq 0$. Let $|b| = \epsilon$. Then by Archimedean Property (2) $$\exists n^* \in \mathbb{Z}^+…
jvargas
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Compactly supported bounded functions with zero integral are dense in Lp spaces

As in the title, I want to know if compactly supported bounded functions with zero mean, i.e. $\int f dx= 0$, are dense in $L^p$ for any $ 1
Lim
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Metrics on Euclidean spaces

I vaguely remember a theorem that says that any two metrics on the Euclidean space $\mathbb{R}^n$ are equivalent in some sense, but probably not in the sense of metric equivalence: two metrics $d_1$ and $d_2$ are said to be metrically equivalent if…
Michael C
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How do i prove that $C_c(X)$ is a vector space?

Let $X$ be a topological space and $C_c(X)$ be the set of all continuous complex functions on $X$ whose support is compact. Let $f,g\in C_c(X)$. Trivially, $f+g$ are continuous, but how do i prove that supp$(f+g)$ is compact?
Katlus
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$f:(0,\infty)\rightarrow (0,\infty)$ is continuous, $\lim_{n\rightarrow\infty}f(nx)=0$ then $\lim_{x\rightarrow\infty}f(x)=0$

Could any one tell me how to solve this one? $f:(0,\infty)\rightarrow (0,\infty)$ is continuos, $\lim_{n\rightarrow\infty}f(nx)=0$ then $\lim_{x\rightarrow\infty}f(x)=0$ I have no clue how to solve this one. Thank you.
Myshkin
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What's the relationship between interior/exterior/boundary point and limit point?

I'm learning real analysis. I found that there are two classification of points: interior/exterior/boundary point and limit point. What's the relationship between interior/exterior/boundary point and limit point ?
Jill Clover
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Is my solution correct to prove that $f (x) = 0$ for all $x \in [a, b]$?

Possible Duplicate: $f\geq 0$ and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$ Is this a correct solution? Thanks for your help
Miguel Mora Luna
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Open covers for $\mathbb{R}$

Does it seem plausible to conjecture that given an enumeration of the rational numbers $\{ q_1, q_2, q_3, \ldots \}$ and a convergent sequence $\{ \epsilon_n \}_{n \in \mathbb{Z^+}} \subseteq \mathbb{R^+}$ such that $\lim \epsilon_n = 0$ but…
brandao
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Continuity of $\arg (z)$

Let $\mathrm{arg}:\mathbb{C}\setminus \{0\} \to [0,2\pi) $ be the function with $\mathrm{arg}(re^{i\alpha})= \alpha$ and $\alpha \in [0,2\pi)$. How can we prove that $\mathrm{arg}: \mathbb{C}\setminus A \to \mathbb{R}$ is continuous when $A= \{z\in…
Josh
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a problem on continuity on rational and irrational point

[NBHM_2006_PhD Screening Test_Topology] Let $f$ be the function on $\mathbb{R}$ defined by $f(t) = \frac{p+\sqrt{2}}{q+\sqrt{2}}−\frac{p}{q}$ if $t = \frac{p}{q}$ with $p, q \in \mathbb{Z}$ and $p$ and $q$ coprime to each other, and $f(t) = 0$…
poton
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How to maximize this integral?

Rudin asked me to maximize $$\int^{1}_{-1}x^{3}g(x)dx$$ under the restraint that $$\int^{1}_{-1}g(x)=\int^{1}_{-1}xg(x)dx=\int^{1}_{-1}x^{2}g(x)dx=0$$ This is clearly a Hilbert space problem need to use orthogonal relations. I computed $x^{3}$'s…
Bombyx mori
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Prove $\frac{1}{x^2}$ is uniformly continuous on $[1,\infty)$ but not on $(0,1)$.

Prove $\frac{1}{x^2}$ is uniformly continuous on $[1,\infty)$ but not on $(0,1)$. Proof On $[1,\infty)$: $$\left|f(x) - f(y)\right| = \left| \frac{1}{x^2} - \frac{1}{y^2} \right| = \frac{(x+y)\left|x -y\right|}{x^2y^2} = \left(\frac{1}{xy^2} +…
Zduff
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