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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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find the condition that minimum of metrics is a metric

suppose $X$ is a set and $e$, $f$ are two metric on the set $X$. Then I knew that $$(x,y)\rightarrow \min\{e(x,y),f(x,y)\}$$ is not a metric on $X$. There are counterexamples in which triangle inequality fails. But I want to know under what…
Yogi
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Proof in Bartle's book

I am studying real analysis by my own and stumble into the following theorem, that Bartle's proves in its appendix, however, I do not understand the case 2. Specifically I do not understand why does he define $h_1$, and the induction hypothesis is…
Leonhard Leibniz
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Compactness properties imply continuity

Please help me do the following. Suppose that $f:\mathbb{R}^m\to\mathbb{R}$ satisfies two conditions: (i) For each compact set $K$, $f(K)$ is compact. (ii) For any nested decreasing sequence of compacts $(K_n)$, $$f\left(\bigcap…
stargitmel
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Is there an example of bounded non-measurable function?

My teacher's lecture note states bounded function defined on a measurable set is not necessarily measurable. Can anyone help give a concrete example? Thank you!
Sherry
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Proving that $\ln ^3|x|=x$ has exactly 3 real solutions

Prove that $$\ln ^3|x|=x$$ Has exactly 3 real solutions. So far my idea is to look separately at $x>0$ and $x<0$ but I'm still stuck with the former. Let $$f(x)=\ln^3|x|-x$$ Assuming $x>0$, as $\lim_{x\to0^+}=-\infty$ and…
Nescio
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Can we use Rolle's Theorem if $f : [a, b] \to \mathbb{R}$ is continuous on $(a, b)$ only?

Why in Rolle's theorem the function is given to be continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a,b)$? What if we take open interval for continuity as well? Please answer if anyone knows.
Waqar Ali Shah
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Mean value theorem for integration in two dimensions

The mean value theorem for integration says that, if $G$ is a continuous real-valued function defined over an interval, $G: [a,b] \to \mathbb{R}$, then the mean value of G on the interval is achieved as a certain point of the interval,…
Erel Segal-Halevi
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Keen Non-Measurable Set - Well Known?

There's a "construction" of a non-measurable set that seems very keen to me; wondering whether anyone's seen it before. Tedious part: For $n,j\in\mathbb Z$ define the dyadic interval $I_{n,j}$ by $$I_{n,j}=[j2^{-n},(j+1)2^{-n}).$$ If $I$ is a…
David C. Ullrich
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Infinite area but less than $1/x$ (eventually)

This question has been bugging me for the past few days. I suspect that it is too difficult for my level of maths (3rd year undergrad). I have enjoyed struggling and wrestling with it but now I must get back to exam revision...I give up. I think I…
Adam Rubinson
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Monotone Functions on R are measurable. What about multidimensional functions? General ordered metric Spaces?

Any function $\mathbb{R} \to \mathbb{R}$ that is monotone is measurable. How far does this generalize? Is a monotone function $\mathbb{R}^n \to \mathbb{R}$ measurable? A function from one totally ordered metric space to another?
Nahpetz
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$\ln(1+x)$ -- why does its power series converge for $|x| < 1$?

Is it a similar reasoning for the convergence of a geometric series, when $|x|<1$? ...at $-1$, $\log(1+x) = \log(0)$, which is undefined. For $x < -1$, $\log(1+x)$ is negative, which is also undefined. So, the left hand inequality of $|x|< 1$ makes…
user243970
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Importance of compactness in Rudin problem.

Okay the problem goes like this: Suppose X, Y, Z are metric spaces, and Y is compact. Let $f$ map X into Y, let g be a continuous one-to-one mapping of Y into Z, and put $h(x)=g(f(x))$ for $x\in X$. Prove that $f$ is uniformly continuous if h is…
user160110
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Baby Rudin Theorem 3.7 Clarification

$\bf 3.7\ \ $ Theorem $\ \ $ The subsequential limits of a sequence $\{p_n\}$ in a metric space $X$ form a closed subset of $X$. Proof $\ \ $ Let $E^*$ be the set of all subsequential limits of $\{p_n\}$ and let $q$ be a limit point of $E^*$. We…
Alain
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Proof that a number uniquely determines its multiplicative inverse

I'm currently working through Pugh's Analysis (for fun.) Currently, I'm working on the following problem: A multiplicative inverse of a nonzero cut $x=A|B$ is a cut $y=C|D$ such that $x*y=1^*$. If $x>0^*$, what are $C$ and $D$? If $x<0^*$, what are…
user17137
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Showing a sequence of functions converges uniformly

Let $\{f_n\}$ be a decreasing sequence of continuous functions with $f_n:[0,1]\rightarrow\mathbb{R}$, with the property that there exists an $M\in(0,1)$ such that $|f_n|\leq M$. Furthermore, $f_n\rightarrow f$ pointwise, where $f$ is also…
Daniel Montealegre
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