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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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Incorrect logic in popular proof of the irrationality of $\sqrt2$?

A popular proof of the irrationality of $\sqrt2$ is to first assume that the number is rational. This means that $\sqrt2=a/b$ where $a$ and $b$ are integers. Another assumption is that $a$ and $b$ are coprime. It turns out that this leads to a…
Jostein Trondal
  • 378
6
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2 answers

Every dense $G_\delta$ subset of $\Bbb R$ is uncountable

Every dense $G_\delta$ subset of $\Bbb R$ is uncountable. I know that I have to use Baire's Theorem but I don't know how. Thanks!
user113576
  • 572
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Real numbers via equivalence classes of Cauchy sequences and the Completeness Axiom

Hi so there's a question in my elementary real analysis course I'm a little bugged about. The question goes: Use the definition of the real numbers via equivalence classes of Cauchy sequences to prove the Completeness Axiom. I tried using proof by…
Oscar Flores
  • 63
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Archimedian field $K$ has LUB property iff it's complete requires DC?

The Setting Let $K$ be an Archimedean field. TFAE: $K$ has the least upper bound property. Every Cauchy sequence in $K$'s additive group converges. Now proving that 1 implies 2 is easy, but the other direction is slightly harder. Not that that's a…
kahen
  • 15,760
6
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2 answers

Can f not be in L1 if its Fourier transform is in L-infinity?

Can $f\notin L^1$ if its Fourier transform $\hat f \in L^\infty$ ? This is a question about the endpoint of Pontryagin duality. We know that if a function is in $L^1$, then its Fourier transform lies in $L^\infty$. This is very easy to show. But…
Gandhi Viswanathan
  • 384
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3 answers

How to properly construct an $\epsilon-N$ proof

I've asked a couple of questions now on this type of proof. My question is: could someone give me a step-by-step set of steps to follow in the general case (e.g. for proof by induction, I'd say the 3 main steps are: check the statement's true in the…
beep-boop
  • 11,595
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1 answer

Open $\sigma$-compact sets with finite measure

Let $X$ be locally compact Hausdorff space and let $\mu$ be positive Borel measure, finite on compacts, outer regular with respect to open subsets, for each Borel set, and inner regular with respect to compact subsets, for each open set and for each…
Richard
  • 4,432
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Why the set of discontinuities can't be uncountable

I am trying to understand why the set of discontinuities of an increasing function $f: \mathbb R \to \mathbb R$ must be finite or countable. I showed that such function can only have jump discontinuities. It is not clear to me why the…
blue
  • 2,884
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Smoothness of a real-valued function on $\mathbb{R}^n $

Let $$ f(x)= \begin{cases} \exp\left(\frac{-1}{1-|x|^2}\right), &\text{ if } |x| < 1, \\ 0, &\text{ if } |x|\geq 1. \end{cases} $$ Prove that $f$ is infinitely differentiable everywhere. ($x$ belongs to $\mathbb{R}^n$ for fixed $n$.) Well,…
somesaymaths
  • 111
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2 answers

Exercise on Boolean closure of elementary sets

I am reading Tao's notes on measure theory and I am stuck with an exercise, so here is the problem: Definition We define $|I|$ the lenght of an interval of endpoints $a
user100106
  • 3,493
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2 answers

Real analysis derivate problem. How can I prove this?

Let $f$ be differentiable in $[a,b]$ and $f(a)=0$. If $\exists M \in R$ such that $\vert f'(x) \vert \leq M \vert f(x)\vert \ $, $\forall x \in [a,b]$, then $f(x)=0, \ \forall x \in [a,b]$. $\\$ I have an idea, but I can´t see it clear. I have…
guerraufo
  • 135
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Bounded variation and differentiability implies absolute continuity

Problem: Let $f$ be a real-valued function everywhere differentiable function on $[0,1]$. If $f$ is of bounded variation on $[0,1]$, then it is absolutely continuous on $[0,1]$. This follows from the Banach–Zaretsky theorem, the mean value theorem,…
Zed
  • 106
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Does convergence on finite intervals imply uniform convergence?

Let $f_n(x) \rightarrow 0$, $n\rightarrow \infty$ for all $x \in \mathbb{R}$. Does this imply $f_n(x) \rightarrow 0$ uniformly on finite intervals? I could think it could be proofen like this maybe: Let the interval $I$ be compact without loss of…
Johannes L
  • 699
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0 answers

$f \in L^1(- \infty , + \infty)$ , Find the limit $ \lim_{n \to \infty} \int^{{+ \infty}}_{- \infty} \frac{f(x)e^{nx}}{1+e^{nx}} dx $

$f \in L^1(- \infty , + \infty)$ , Find the limit $ \lim_{n \to \infty} \int^{{+ \infty}}_{- \infty} \frac{f(x)e^{nx}}{1+e^{nx}} dx $ My Attempt $$ \lim_{n \to \infty} \int^{{+ \infty}}_{- \infty} \frac{f(x)e^{nx}}{1+e^{nx}} dx = \lim_{n \to…
the8thone
  • 4,111
6
votes
1 answer

Inverse rule to the L'Hôpital's rule

If in L'Hôpital's rule we have that: $f,g :(a,b)\to \mathbb{R}$, there exist $f'(x)$, $g'(x)$, and $g'(x)\ne0$, $$\lim_{x\to a^+} \frac{f(x)}{g(x)} = L,$$ and also $\lim_{x\to a^+} f(x)=\lim_{x\to a^+} g(x) = 0$, must it be also $$\lim_{x\to…
Steel_Rat11
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