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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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Triangle inequality and the square root of a metric space

for (i) I know that the square root part is true but I don't know how to put it into words to prove it. For (ii) I just don't know how top apply the requirements for a metric space to the square root of another metric space. Just kind of confusing…
Daniel
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Exercise 3.9 in rudin's real and complex analysis

Problem Suppose $f$ is Lebesgue measurable on $(0,1)$ and not essentially bounded. By Exercise $4(e)$ $\|f\|_{p}\to \infty $ as $p\to \infty $.Can $\|f\|_{p}$ tend to $\infty $ arbitrarily slowly? More precisely ,is it true that to every positive…
pxchg1200
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How discontinuous does a $[a,b] \to (c,d)$ bijection have to be?

I know that bijection from $[a,b]$ to $(c,d)$ can't be continuous, but I'm wondering if such a function could exist if it was discontinuous at just countable points, and if this isn't possible, why?
YoTengoUnLCD
  • 13,384
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Continuity in two dimensions

How would you prove or disprove that the function given by $$f(x,y) = \begin{cases} \frac{x^3y^2}{x^4 + y^4} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$ is continuous at $(0,0$). I tried to think of a function where the limit approached…
user26069
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a question of theorem 3.13 in Real and Complex Analysis, Rudin

Here is the Theorem 3.13 in rudin's real and complex analysis book， Theorem 3.13 Let $S$ be the class of all complex,measureable,simple functions on $X$ such that \begin{equation} \mu(\{x:s(x)\neq 0\})<\infty \tag{1} \end{equation} If $1\leq…
pxchg1200
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Prove $|a|=\sqrt{a^2}$

I am having difficulty showing that the last case holds and I also just wanted to make sure I am proving this correctly. Case (i): If $a=0$ then we have $|a|=0=\sqrt{0^2}$ so trivially this is true. Case (ii): If $a>0$ then we have…
Craig
  • 652
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Prove that there is a smallest point in the interval $[0,1]$ at which the function $f$ attains the value $0$

Suppose that the function $f:[0,1]\rightarrow \mathbb{R}$ is continuous, $f(0)>0$, and $f(1)=0$. Prove that there is a number $x_0 \in (0,1]$ such that $f(x_0)=0$ and $f(x)>0$ for $0\le x < x_0$; that is, there is a smallest point in the interval…
Simple
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Isometry in $\mathbb{R}^n$

I'm trying to prove that if $f\colon\mathbb{R}^n \to \mathbb{R}^n$ is a $\mathcal{C}^1$ mapping such that $f'(x)$ is a (linear) isometry for every $x \in \mathbb{R}^n$, then $f$ is an isometry. By an application of inverse mapping theorem and mean…
Daniel
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Proper Measurable subgroups of $\mathbb R$

If $(\mathbb{R},+)$ is a group and $H$ is a proper subgroup of $\mathbb{R}$ then prove that $H$ is of measure zero.
anonymous
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Product $\sigma$-algebra in countable case (Proposition 1.3 in Folland)

In the real analysis, by Folland, p. 23: I know $\prod_{\alpha\in A}E_{\alpha}=\bigcap_{\alpha\in A}\pi_{\alpha}^{-1}(E_{\alpha})$. But I cannot figure out why the product $\sigma$-algebra in the countable case should be defined in…
sleeve chen
  • 8,281
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Suppose $\lim \sup_{n \to \infty}a_n \le \rho$. Show $\lim \sup_{n \to \infty} a_n^{{(n-m)}/{n}} \le \rho$.

Suppose that $\{a_n\}$ is a sequence of positive numbers with $\lim \sup_{n \to \infty} a_n\le \rho$. Show $\lim \sup_{n \to \infty} a_n^{{(n-m)}/{n}} \le \rho$, where $m \in \Bbb N$. I was thinking about doing something along the lines of $\lim…
Meecolm
  • 683
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Riemann Integration Problem

Let $f$ be a continuous function on $[0,1]$ satisfying $$\int_0^1f(x)\,dx = 0$$ and $$\int_0^1xf(x)\,dx = 0.$$ Show that there exists $a$,$b$ in $[0,1]$ with $a < b$, such that $f(a) = 0 =f(b)$. Existence of one point is clear to me but I cannot…
Ester
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Prove that $\lim(x_n)=0$ if and only if $\lim(|x_n|)=0$.

Prove that $\lim(x_n)=0$ if and only if $\lim(|x_n|)=0$. Definition: Let $X = (x_n)$ be a sequence in $\mathbb{R}$ and let $x\in\mathbb{R}$. Then $\lim(x_n) =x$ iff for all $\varepsilon>0$, $\exists k\in\mathbb{N}$ such that $|x_n-x|<\varepsilon$…
flubsy
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Smooth extension of a continuous function on the boundary of a domain

Let $\Omega$ be a open, bounded set in $\mathbb{R}^n$. Suppose $g$ is a continuous function defined on the boundary $\partial \Omega$. Then, is it possible to show that there exists a function $f$ defined (and continuous) on $\Omega \cup \partial…
user74261
  • 2,286
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caratheodory measurability condition

(1) If $E$ is a subset of $\mathbb R$ with finite outer measure, i.e. $m^{*}(E) <\infty$; and (2) $E$ is not Lebesgue measurable, i.e. there exists $F$ such that $m^{*} (F) < m^{*}(EF) + m^{*}(FE^{c}).$ [Claim] There exists an open set $O\supset…
user79963
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