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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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Continuous bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$

Question is to check which of the following holds (only one option is correct) for a continuous bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$. $f$ has to be uniformly continuous. there exists a $x\in \mathbb{R}$ such that $f(x)=x$. $f$ can…
user87543
6
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2 answers

Cantor construction is continuous

I define a function $f:\mathbb{R}\to\mathbb{R}$ as follows: $f(x)=0$ for $x\le 0$. $f(x)=1$ for $x\ge1$. $f(x)=\dfrac12$ for $x\in\left[\dfrac13,\dfrac23\right]$. $f(x)=\dfrac14$ for $x\in\left[\dfrac19,\dfrac29\right]$, $f(x)=\dfrac34$ for…
Kunal
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Prove subset of $\mathbb R^2$ is open

Problem. Let $G = \{(x,y): x \ne y\}$. Prove $G$ is an open subset of $\mathbb R^2$. What I am thinking: If I could show that $\mathbb R^2 \setminus G = \{(x,y): x = y\}$ is a closed set, then its complement $G$ is open. I might be totally off. Any…
MDW
  • 366
6
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limit for applying of sin operation n times and multiplying the result by square root of n

Numerically, I found that if one builds a sequence of $sin(sin(sin(....(sin(x)...))$ with $n$ being the number of times sin operation is performed, then with n going to infinity the product of this operation multiplied by the square root of n…
user36225
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Show that $T$ is or isn't a strict contraction but have fixed points

Define $T: C[0,1] \to C[0,1]$ by $$(Tf)(x) = \int^x_0 f(t)dt.$$ Show that $T$ is not a strict contraction while $T^2$ is. What is the fixed point of $T$?
user10311389
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Can we divided the irrationals into a countable disjoint union of subsets, none of which has a rational limit point?

If we partite $\mathbb{R}\backslash\mathbb{Q}$ as $\cup_{i\in\mathbb{N}}A_i=\mathbb{R}\backslash\mathbb{Q}$, $A_i\cap A_j=\emptyset$ if $i\ne j$, can it hold that $A_i$ has no rational limit point? Actually, we have irrational perfect sets, but…
Anonymous Coward
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Smoothness of $f(\sqrt{x})$

How to prove that if a function $f\colon\mathbb R\to\mathbb R$ is of the class $C^{2n}$ and even then there exists a function $g\colon\mathbb R \to\mathbb R$ of the class $C^n$ such that $f(x)=g(x^2)$. It is obvious, by using power series expansion…
Richard
  • 4,432
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3 answers

Show that $f(x)=\sqrt{x}:[0,1]\rightarrow \mathbb{R}$ is not a Lipschitz function.

A function $f:D\rightarrow \mathbb{R}$ is said to be a Lipschitz function provided that there is a nonnegative number $C$ such that $|f(u)-f(v)|\le C|u-v|$ for all $u,v\in D$ We want to show there are there exist $u,v\in [0,1]$ such that…
TheMobiusLoops
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2 answers

Continuity implies Borel-measurability?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. Is it necessary that $f$ is Borel-measurable? I'm considering $A=f^{-1}((a,\infty))$ where $a\in\mathbb{R}$. Is $A$ necessarily a Borel set? It looks like it should be, but I'm not…
Mika H.
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2 answers

Solution to integral curve for compactly supported functions

Let $U\subseteq\mathbb{R}^n$ be an open subset, and let $g:U\rightarrow\mathbb{R^n}$ be a $C^1$ function. Let $x_1(t),\ldots,x_n(t)$ be $C^1$ functions on an open interval $I\subseteq\mathbb{R}$. Write $x(t)=(x_1(t),\ldots,x_n(t))$. Consider the…
PJ Miller
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2 answers

Locally Lipschitz does not imply $C^1$

Let $A$ be open in $\mathbb{R}^m$; let $g:A\rightarrow\mathbb{R}^n$. If $S\subseteq A$, we say that $S$ satisfies the Lipschitz condition on $S$ if the function $\lambda(x,y)=|g(x)-g(y)|/|x-y|$ is bounded for $x\neq y\in S$. We say that $g$ is…
Mika H.
  • 5,639
6
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4 answers

how to show strictly increasing function on an interval has continuous inverse

my question is to show that a strictly increasing function that is defined on an interval has a continuous inverse. Thanks.
Yang
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How to show that a limit vanishes

Let $1
user1736
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1 answer

Continuous Function

Let $ f: \Bbb R^2 \to \Bbb R $ such that : $ \forall _{y_0 \in \Bbb R\ }: $ function $ x \to f(x,y_0) $ continuous function and increasing $ \forall _{x_0 \in \Bbb R\ }: $ function $ y \to f(x_0,y) $ continuous function I mean the continuity of…
P10D
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