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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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Arbitrary Union and Intersection of Closed and Open Sets

I have four quick questions and have listed them below. I am seeking for corroboration of the first three and a bit of insight on the fourth, as I have hit a solid brick wall. Definition. A set $A\subseteq\mathbb{R}$ is called an $F_\sigma$ set if…
wjmolina
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Question about differentiating under the integral sign

Let $01$, and define the function $F:[0,\infty) \to [0, \infty)$ as $$F(t)= \int_E [f(x)+tg(x)]^pd x.$$ Show $F$ is…
Darrin
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Distance is (uniformly) continuous

I've been told that $$d(x,A) = \inf_{y \in A} |x-y|$$ is uniformly continuous, but I don't understand why? Is there a short proof of this statement or is this a slightly deeper result? This was a result discussed in my analysis lecture.
dalastboss
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How to prove that $f(x)=x^{x^x}$ is strictly increasing without calculating the derivative?

How to prove that $f:(0,\infty )\to\mathbb R$ defined by $f(x)=x^{(x^x)}$ is strictly increasing without calculating the derivative?
idm
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Integral of bounded continuous function on $R$

Let $f$ be a bounded continuous function on $R$. Prove that $$\lim_{n \to \infty} \frac{n}{\pi} \int_{ R} \frac {f(t)}{1+n^{2}t^{2}} dt=f(0)$$ I solved this question as follows, but I ran into a problem: Solution: Since $f$ is continuous at $0$,…
M65
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$\mathbb{Q}^2$ is dense in $\mathbb{R}^2$

I know that $\mathbb{Q}$ is dense in $\mathbb{R}$. What is the next step to prove that $\mathbb{Q}^2$ is also dense in $\mathbb{R}^2$? Any hints are welcomed.
Sarah. N
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Equicontinuity and modulus of continuity

A modulus of continuity for a function $f$ is a continuous increasing function $\alpha$ such that $\alpha(0) = 0$ and $|f(x) - f(y)| < \alpha(|x-y|)$ for all $x$ and $y$. I am trying to prove that an equicontinuous family $\mathcal F$ of functions…
user15464
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Is the derivative of a differentiable function continuous a.e.?

Let $f:[a,b]\rightarrow \mathbb{R}$ be a differentiable function. I know that $f'$ does not need to be continuous on $[a,b]$. However, all counterexamples I know has finite discontinuities. I want to know whether $f'$ is continuius a.e. on $[a,b]$.…
Mathemagic
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Showing $\ln(\sin(x))$ is in $L_1$

Prove $\ln[\sin(x)] \in L_1 [0,1].$ Since the problem does not require actually solving for the value, my strategy is to bound the integral somehow. I thought I was out of this one free since for $\epsilon > 0$ small enough, $$\lim_{\epsilon \to…
Darrin
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Dyadic Rational are dense

I want to prove that set of all dyadic rational numbers in $[0,1]$ is dense in $[0,1]$ but I do not want to prove it using binary expansion of a number. Is there any other proof for the same?
Abcd J
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Separability $\mathbb{R}^2$

I know that $\mathbb{R}^2$ with the post office metric is not separable. And the post office metric is defined by $m(x,y) = d(x,0) + d(y,0)$ for distinct points $x$ and $y$, and $m(x,x) = 0$ where $d$ is metric on $\mathbb{R}^2$. My idea for the…
marvinthemartian
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Is there some consensus on the dimensions of a Jacobian matrix and of a gradient?

According to Wikipedia, given a differentiable mapping $F: \mathbb{R}^n \to \mathbb{R}^m$, its Jacobian matrix is a $m \times n$ matrix defined as: $$ J_F=\begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \cdots & \dfrac{\partial y_1}{\partial…
Tim
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If a function is uniformly continuous on $(-\infty,-1]$ and $[-1,\infty)$ is it uniformly continuous on $\mathbb{R}$

Is it correct to say that if $f(x)$ is uniformly continuous on $(-\infty,-1]$ and $[-1,\infty)$, then it is uniformly continuous on $\mathbb{R}$? I don't think this is true but cannot think of a counterexample. Below there is an example of where I…
hmmmm
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Showing that $f(x)=e^{-x}$ is uniformly continuous on $[0,\infty)$

I am trying to show that $f(x)=e^{-x}$ is uniformly continuous on $[0,\infty)$ and not having much success. I'm attempting to use a modified version of the following result (which I found on Math Stack Exchange here:…
mrmingus
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Absolute continuity and sets of measure 0

I have a problem with a statement in Rudin's book "Real and Complex Analysis" (3rd edition) - proof of Theorem 7.18 Let $f:[a,b] \mapsto \mathbb{R}$ continuous non decreasing. If $f$ maps sets of measure $0$ to sets of measure $0$, then the…
kkloo
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