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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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Clarification requested about a claim in Abbott's "Understanding Analysis"

In the opening sentences of chapter 3.5 on Baire's Theorem, Abbott writes: The structure of open sets is relatively straightforward. Every open set is either a finite or countable union of open intervals. I have looked backed through chapter 3 and…
dnlwng
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Showing image of an integral operator is continuous

Let $K(x,y)$ be continuous on the unit square $[0,1] \times [0,1]$ and let $\| K \| = \max | K(x,y) |$. For $\phi (x) \in C([0,1])$, define $$T\phi (x)= \int_{0}^{1}K(x,y)\phi(y)dy$$ (a) Show that $T\phi \in C([0,1])$ , i.e. $T: C([0,1]) \to…
James McCoy
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Values for which tetration to infinite heights (i.e., $x^{x^{x^{x^{.^{.^{.}}}}}}$) converges

I was reading Evaluating tetration to infinite heights (e.g., $2^{2^{2^{2^{.^{.^.}}}}}$). For what values of $x$ does tetration to infinite heights (i.e., $x^{x^{x^{x^{.^{.^{.}}}}}}$) converge?
GA316
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A question about nonmeasurable subset of real line and uncountable measure-zero subset.

When I was reviewing real analysis, I have a question below. Does every nonmeasurable subset of real line have an uncountable and measure-zero subset? Just consider the Lebesgue measure. I have known that this is true for every measurable set…
Mod.esty
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Proving pointwise convergence given that $f$ is bounded and cont

Given $f:X\to\mathbb{R}$ be a bounded function such that $\exists M$ s.t $|f(x)|\leq M$ and $f_k:X\to\mathbb{R}$ where $f_k=\inf_{y\in X}(f(y)+kd(x,y))$ Let $k\in\mathbb{N}$ and $X=\mathbb{R}$ and $d(x,y)$ be the usual metric. Let $(X,d)$ and $f$ be…
Undergrad2019
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Proving $f(x)=\begin{cases}0&, x\in\mathbb Q\\x^2&,x\in\mathbb R\setminus\mathbb Q\end{cases}$ is differentiable only at $x=0$

Consider the function $$f(x)=\begin{cases}0&,x\in\mathbb Q\\x^2&, x\in\mathbb R\setminus\mathbb Q\end{cases}$$ Prove that $f$ is differentiable only at $x=0$. My approach: Let us take any point $a$ and assume that $f$ is continuous at $a$. We…
Sanket Biswas
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Is $|x|^{3/2}$ differentiable?

Given $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = |x|^{3/2}$ Then choose the correct option $1.$ $f$ is differentiable $2.$ $f$ is differentiable but not continuously differentiable My attempt: I think option $2$ is correct, i.e.,…
jasmine
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Injectivity and Surjectivity of the Exponential Function

Why is the exponential function injective but not surjective?
Mike Oliver
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Proving differentiability without using the closed-form integration

It is well known that $F(y) = \int_0^\infty \frac{\cos(xy)}{1+x^2}dx = \frac{\pi e^{-y}}{2}$ for $y \geqslant 0$. Without knowing this a priori I want to show that $F(y)$ is differentiable on $(0,\infty)$ and that the right-hand derivative $F'_R(0)$…
WoodWorker
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Real analysis: continuity of functions

Is it possible for an increasing function $[0,1]$ to be discontinuous at every irrational number? Can you help me with this problem?
birzh
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If $(\limsup_{n\to\infty} x_{n})\cdot(\limsup_{n\to\infty} 1/x_{n})=1$, then the sequence $x_n$ converges

${x_{n}}$ is a sequence in $\mathbb{R}$ where $n\geq 1$ such that all $x_{n} >0$. If $(\limsup_{n\to\infty} x_{n})\cdot(\limsup_{n\to\infty} 1/x_{n})=1$, how can I show that ${x_n}$ converges? I am thinking about using the contrapositive of the…
user711033
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Show that there is at least one $\varepsilon\in(0,1)$, such that $ f(x)=-1+x^2+\frac{x^2(x-1)}{3!}f'''(\varepsilon),\quad x\in(0,1). $

Suppose $f(x)$ is three-times differentiable on $[0,1]$, and $f(0)=-1,\;f(1)=0,\;f'(0)=0$. Show that for any $x\in(0,1)$ there is at least one $\varepsilon\in(0,\,1)$, such that $$ f(x)=-1+x^2+\frac{x^2(x-1)}{3!}f'''(\varepsilon). $$ The Taylor…
Knt
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Given $x_{0}$, let $f(x) = \|x-x_{0}\|$. Show tha $f$ has a minumum on any closed, nonempty set $A \subset \mathbb{R^{n}}$

Given $x_{0}$, let $f(x) = \|x-x_{0}\|$. Show tha $f$ has a minumum on any closed, nonempty set $A \subset \mathbb{R^{n}}$. I tried to do a test for reduction to the absurd, but it was a little difficult to get the result. Perhaps you find a simpler…
Curious
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an infinite product identity

I encountered the following post on a website (www.quora.com) I quote the post verbatim-these are not my comments. "Why is the number "42" so significant to mathematicians? It is for the very simple reason…
student
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