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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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Increasing $g$ where $g' = 0$ a.e. but $g$ not constant on any open interval?

As the question title suggests, does there exist an increasing function $g$ such that $g' = 0$ almost everywhere but $g$ isn't constant on any open interval?
user378405
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Closed set in $\mathbb{R}^n$ is closure of some countable subset

Let $A$ be a closed set in $\mathbb{R}^n$ . How can I show that $A$ = closure of $B$ where $B$ is countable ? Thanks for any help .
Ester
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Proving $f(x+a) \geq f(x)$ almost everywhere

I encountered the following problem in past analysis qualifying exam: Problem. Let $f \in L_{loc}^{1}(\mathbf{R})$ be real valued and assume that for each $n > 0$, we have $f(x+ \frac{1}{n}) \geq f(x)$ for almost all $x \in \mathbf{R}$. Show that…
Younghak Kwon
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Video Lectures that closely follow Rudin's Real Analysis/Royden's Real Analysis

I plan to apply for the financial engineering course at NTU, Singapore. I intend to study real analysis and measure theory, as I have the mornings and evenings to myself after work. I am working through Rudin's PMA currently, with the help of online…
Quasar
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Show that there does not exist a strictly increasing function $f : \mathbb Q \to \mathbb R$ such that $f(\mathbb Q) = \mathbb R$.

The following exercise in an analysis text and I am trying to solve it without concepts of general topology but fail. Show that there does not exist a strictly increasing function $f : \mathbb Q \to \mathbb R$ such that $f(\mathbb Q) = \mathbb R$.…
user200918
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what functions or classes of functions are Riemann non-integrable but Lebesgue integrable

I am wondering if there are some other examples of Riemann non-integrable but Lebesgue integrable, besides the well-known Dirichlet function. Thanks.
Qiang Li
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Proof of uniform continuity of $\frac{1}{x}$

Show that the function $f(x) = \frac{1}{x}$ is not uniformly continuous on the interval $(0,\infty)$ but is uniformly continuous on any interval of the form $(\mu, \infty)$ if $\mu > 0$. My Work Referring to the definition of uniform continuity, I…
Moderat
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Show that $A=\{ \frac{m}{2^n}:m\in \mathbb {Z},n\in \mathbb {N} \} $ is dense in $\mathbb {R}$

Possible Duplicate: Density of a Set on $\mathbb{R}$? I have to show that show that $A=\{ \frac{m}{2^n}:m\in \mathbb {Z},n\in \mathbb {N}\} $ is dense in $\mathbb {R}$. A set A is dense in $\mathbb {R}$ if $\overline A=\mathbb {R}$. But also $Y$…
HipsterMathematician
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Intersection of infinite sets is infinite?

I know that if $C \subseteq [0,1]$ is uncountable, then there exists $a \in (0,1)$ such that $C \cap [a,1]$ is uncountable. Is it still true for any infinite sets? That is, if $C \subseteq [0,1]$ is infinite, does there exist an $a \in (0,1)$ such…
mmarky
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Stone-Weierstrass theorem proof (Rudin)

I am reading Rudin's proof (3rd edition) and am wondering what substitution is made to make it true that $P_n(x)=$ the integral from $-x$ to $1-x$ is equal to the same function integrated from -1 to 1. He says there's a substitution but I haven't…
Geakee1
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Deleted versus non-deleted limits

When I read Serge Lang's Undergraduate Analysis(second edition) page 41, I found the definition of limit he define is so called non-Deleted limits, this results in some conclusion is different from the usual limit( deleted limit), for example,…
noname1014
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What exactly is an "analytic function"?

This is for real analysis so I'm not worried about complex analytic functions. The definition in my book just says: "A function f(x) which is represented by a power series with a positive radius of convergence is said to be 'real analytic at the…
bones_mccoy
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Graph of a continuous function has measure zero

I need help to solve the following problem: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous function. Prove that the graph $G(f)=\{(x,f(x)):x\in\mathbb{R}^n\}$ has measure zero in $\mathbb{R}^{n+1}$. I suppose that I have to use that f…
user326159
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function is smooth iff the composition with any smooth curve is again smooth

I'm stuck on the following part of a proof: Let $\phi: \mathbb R^m \to \mathbb R^n$ be a function such that $\gamma'(t) := \phi(\gamma(t))$ is smooth for every smooth function $\gamma: \mathbb R \to \mathbb R^m$. I want to show that $\phi$ is…
Sam
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Which of the following are Dense in $\mathbb{R}^2$?

Which of the following sets are dense in $\mathbb R^2$ with respect to the usual topology. $\{ (x, y)\in\mathbb R^2 : x\in\mathbb N\}$ $\{ (x, y)\in\mathbb R^2 : x+y\in\mathbb Q\}$ $\{ (x, y)\in\mathbb R^2 : x^2 + y^2 = 5\}$ $\{ (x, y)\in\mathbb…
preeti
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