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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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Show that a convex compact set in $R^2$ can be cut into 4 sets of equal area by 2 perpendicular lines

Okay I need to show this using calculus and mean value theorem. My try : Let $D$ be a convex and compact set in $R^2$ Now let $R$ be a compact closed rectangle such that $D \subset R$. Draw two lines parallel to the axis such that $D$ is now…
user346936
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Open cover rationals proper subset of R?

If I were to cover each rational number by a non-empty open interval, would their union always be R? It seems correct to me intuitively, but I am quite certain it is wrong. Thanks
Paul
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Is $\{n\pmod \pi: n \in \Bbb N\}$ dense in $[0,\pi]$?

Is $\{n\pmod\pi: n\in\Bbb N\}$ dense in $[0,\pi]$ ? Is there a proof or well known theorem for this result? My intuition would say that it is dense.
ZirconCode
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Can the limit of a sequence of bounded functions be unbounded?

It is assumed that the sequence of functions converges and that each function is bounded, then can the limiting function be unbounded?
DpS
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Equivalence of a Lebesgue Integrable function

I have the following question: Let $X$: $\mu(X)<\infty$, and let $f \geq 0$ on $X$. Prove that $f$ is Lebesgue integrable on $X$ if and only if $\sum_{n=0}^{\infty}2^n \mu(\lbrace x \in X : f(x) \geq 2^n \rbrace) < \infty $. I have the following…
Frank White
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Prove that function has only one maximum

I have a function $f_n(x)$ with an integer parameter $n \in \{3,4,5,\dots\}$ and $x\in ]0,1[$, and i want to show that $f_n(x)$ has only one critical point for every value of $n$. The function is $$ f_n(x)=(1 - x)^n + (1 + x)^n + \frac{ x [(1 -…
kiara
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Countability of set of positive reals with bounded sum for all finite subsets

Consider a set $B$ of positive real numbers such that the sum of elements in any finite subset of $B$ is always less than or equal to $2$. Show that $B$ is countable. I'm trying to find a bijection between $\mathbb{N}$ and $B$ but it's not clear how…
rorty
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Why convolution of integrable function $f$ with some sequence tends to $f$ a.e.

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be integrable with$\int_{\mathbb{R}}g(x)dx=1$ and $|g(x)| \leq \frac{C}{(1+|x|)^{1+h}}$ for $x \in \mathbb{R} $, where $C, h>0$ are constants. Let $g_t(x)=\frac{1}{t} g(\frac{x}{t})$ for $x \in \mathbb{R}$,…
Alex
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Continuity of a function at an isolated point

Suppose $c$ is an isolated point in the domain $D$ of a function $f$. In the delta neighbourhood of $c$, does the function $f$ have the value $f(c)$?
Vinod
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The domain of $x^x$?

This one looks simple, but apparently there is something more to it. $$f{(x)=x^x}$$ I read somewhere that the domain is $\Bbb R_+$, a friend said that $x\lt-1, x\gt0$... I'm really confused, because i don't understand why the domain isn't just all…
Yonatan Izutskiver
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Derivative that doesn't care about countable subsets?

In Lebesgue integration, if you change a function on a countable subset of its domain, neither integrability nor the value of the integral changes. The same is obviously not true for differentiation, which is locally defined through a limit. Now I…
celtschk
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What is the difference between a ring and a $\sigma$-ring?

I started reading Rudin's Principles of Mathematical Analysis for some Lebesgue Theory. Rudin introduces both rings and $\sigma$-rings, but I don't see the difference between them. Assuming I'm not misunderstanding the definition, a ring is a…
Sujaan Kunalan
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one-sided differentiability

Well known theorem: If $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable and $f'(x)=0$ for all $x$, then $f$ is constant. The assumption of differentiability can be weakened to continuity and one-sided differentiability: If…
larry01
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Nowhere continuous real valued function which has an antiderivative

My question: Is there a function $f: \mathbb{R} \rightarrow \mathbb{R}$ that nowhere continuous on its domain, but has an antiderivative? If there is no such a function, is it true to conclude that: to have an antiderivative, $f$ is necessary to…
netsurfer
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Query regarding Theorem 1.21 in Baby Rudin

In the proof of Theorem - 1.21 (pg-10) in Rudin's Principles of Mathematical Analysis (Statement - For every real $x>0$ & every integer $n>0$, there is one & only one positive real $y$ s.t. $y^n = x$); The author says - "Assume $y^n
Ritu
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