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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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What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$?

This question was motivated by another question in this site. As explained in that problem (and its answers), if $\displaystyle f$ is continuous on $\displaystyle [0,1]$ and $\displaystyle \int_0^1 f(x)p(x)dx=0$ for all polynomials $\displaystyle…
Bruce George
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Can a continuous function on [0,1] be constructed which is differentiable exactly at two points in [0,1]?

We see that we can find a function nowhere differentiable or finitely not differentiable. But I want to understand, can a continuous function on $[0,1]$ be constructed which is differentiable exactly at two points in $[0,1]$? How can I construct…
Sayantan Koley
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Is a dense subset of the plane always dense in some line segment?

Consider the following question: Given a dense set in the plane, does there always exist a line segment in which this set is dense? I have been puzzling over this for some time. Can someone help or give me some hints?
nicholas
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Continuity of $\delta$ in the definition of continuity

When I was in the shower this morning a question went through my head about continuity of a function at a point. The simplest formulation of this question is: Let $f : \mathbb{R} \to \mathbb{R}$ be an unbounded continuous function with $f(0) = 0$.…
Clive Newstead
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Prove the set of points where $f$ is differentiable is dense.

Let $I\subset \Bbb{R}$ be an open interval and consider a continuous function $f:I\to \Bbb{R}$ satisfying, for all $x\in I$ $$\displaystyle \lim_{h\to 0} \frac{f(x+h)+f(x-h)-2f(x)}{h}=0$$ Prove that the set of points at which $f$ is…
Mathronaut
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When differentiability of the product implies differentiability of the individual terms?

Say we have $ h(x)=f(x)\cdot g(x)$ where $f$ and $g$ are continuous and strictly increasing. It follows they are differentiable almost everywhere and so is $h$. We also know that $f>0$ and $g>0$. I'm trying to find a straightforward proof that under…
Sergio Parreiras
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Set of Finite Measure: Uncountable disjoint subsets of non-zero measure

Suppose $A$ is a set of finite measure. Is it possible that $A$ can be an uncountable union of disjoint subsets of $A$, each of which has positive measure?
Mykie
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Finite number of zeros?

Is it true that any nonzero function $f: \mathbb R \to \mathbb R$ which is either: 1) constant, or 2) a polynomial, or 3) $\exp$, or 4) $\log$, or 5) any finite combination of the above using addition, subtraction, multiplication, division and…
Roel
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Infinite series of nth root of n factorial

Why is this not correct: $$ \begin{align} \lim_{n\to \infty}\sqrt[n]{n!} &= \lim_{n\to \infty}\sqrt[n]{n(n-1)(n-2)(n-3)\cdots(1)} \\ &=\lim_{n\to \infty}\sqrt[n]{n} \cdot \lim_{n\to \infty}\sqrt[n]{n-1} \cdot \lim_{n\to \infty}\sqrt[n]{n-2}\cdots…
axin
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If $f'$ is differentiable at $a$ then $f'$ is continuous at $(a-\delta,a+\delta)$

Is there a counterexample? Proposition: Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be a differentiable function such that $f':\mathbb{R} \longrightarrow \mathbb{R}$ is differentiable at $a\in\mathbb{R}$ (possibly differentiable at a single…
felipeuni
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Applying analysis to solve a line-of-sight problem

This was an optional h.w. problem: You are at the origin in $\mathbb{Z}\times\mathbb{Z}$. There are trees of a fixed finite radius at each point in $\mathbb{Z}\times\mathbb{Z}$ (other than the origin). How far can you see? This was asked in (I…
user12802
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4 answers

A set is closed if and only if it contains all its limit points.

A set is closed if and only if it contains all its limit points. Proof in book: Suppose $S$ is "not closed". We must show that $S$ does not contain all its limit points. Since $S$ is "not closed", $S^c$ is "not open". Therefore there is at least…
Tom
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Differentiable and Continuous functions on [0,1] with 'weird' conditions.

I've been stuck on this one for a while. Comes from an analysis qual question. Let f be a function that is continuous on $\left[0,1\right]$ and differentiable on $(0,1)$. Show that if $f(0)=0$ and $|f'(x)| \leq |f(x)|$ for all $x \in (0,1)$, then…
DaveNine
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How can I show that $\sup(AB)\geq\sup A\sup B$ for $A,B\subset\mathbb{R}$ where $A\cup B$ is positive and bounded?

The question is based on the following exercise in real analysis: Assume that $A,B\subset{\Bbb R}$ are both bounded and $x>0$ for all $x\in A\cup B$. Show that $$ \sup(AB)=\sup A\sup B $$ where $$ AB:=\{ab\in{\Bbb R}:a\in A, b\in…
user9464
13
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4 answers

Closed subsets of $\mathbb{R}$ characterization

I remember the characterization of open subsets of $\mathbb{R}$ as a countable union of disjoint open intervals. I was thinking about whether this allows us to characterize closed subsets as a countable union of disjoint closed intervals. As we…
PJ Miller
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