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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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Proving $ f(x) = x^2 $ is not uniformly continuous on the real line

This is homework problem and the very premise has me stumped. It's in a text on PDE. The exercise says to show that $ f(x) = x^2 $ is not uniformly continuous on the real line. But every definition I know says that it is a continuous function, and…
Jesse
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A question about Existence of a Continuous function.

Let $f$ be a continuous function on the interval $[1,2]$. It follows from Stone-Weierstrass theorem that if $\displaystyle \int_1^2x^nf(x) \, dx=0$ for integers $n=0,1,2,\ldots$, then $f$ must be identically zero. My question is, Does there exist a…
user70310
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Continuous functions on $[0,1]$ is dense in $L^p[0,1]$ for $1\leq p< \infty$

I tried to show that the continuous functions on $[0,1]$ are dense in $L^p[0,1]$ for $ 1 \leq p< \infty $ by using Lusin's theorem. I proceeded as follows.. By using Lusin's theorem, for any $f \in L^p[0,1]$, for any given $ \epsilon $ $ > $ 0,…
Detectives
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How to show that the set of points of continuity is a $G_{\delta}$

I am trying to solve this exercise from Royden's 3rd edition. The question is as follows: Let $f$ be a real-valued function defined for all real numbers. Show that the set of points at which $f$ is continuous is a $G_{\delta}$. Let $$A_n = \{y :…
Linda
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If $f:\mathbb{R}\to\mathbb{R}$ is a left continuous function can the set of discontinuous points of $f$ have positive Lebesgue measure?

If $f:\mathbb{R}\to\mathbb{R}$ is a left continuous function can the set of discontinuous points of $f$ have positive Lebesgue measure? I wondered this today, but made little progress. Thank you.
Ben Derrett
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Is the image of a Borel subset of $[0,1]$ under a differentiable map still a Borel set?

Let $f:[0,1]\to[0,1]$ be a continuous function such that its derivative $f'$ exists on $(0,1)$. Inspired by a similar question of myself here, I want to ask: If $E\subset[0,1]$ is a Borel set, is $f(E)$ still a Borel set? Remark: It is known…
23rd
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Inverse/Implicit Function Theorem Reasons?

I watched an ICTP lecture on elementary real analysis & the lecturer went to great pains to emphasize the importance of the intermediate value theorem because it is what generalizes to higher dimensions via connectedness & how Bolzano-Weierstrass…
sponsoredwalk
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2 answers

Monotone+continuous but not differentiable

Is there a continuous and monotone function that's nowhere differentiable ?
t.k
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5 answers

Measure of reals in $[0,1]$ which don't have $4$ in decimal expansion

It's an exercise in E. M. Stein's "Real Analysis." Let $A$ be the subset of $[0,1]$ which consists of all numbers which do not have the digit $4$ appearing in their decimal expansion. What is the measure of $A$? I would be grateful if someone can…
molan
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Examples of absolutely continuous functions that are not Lipschitz.

I have just solved an exercise, which asked to show that function $f$ is Lipschitz implies that $f$ is absolutely continuous. However, I'm wondering if the converse is true. I can't seem to think of any counterexamples at the moment. I think I'm…
Linda
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2 answers

Measurability of $\xi$ in the mean value theorem

Suppose $f\in C^1(\mathbb{R})$, by mean value theorem, for any $x\in (0,\infty)$, there exists $\xi(x)\in (0,x)$ such that $$\frac{f(x)-f(0)}{x}=f'(\xi(x)).$$ My question is: Question: Can $\xi(x)$ always be chosen to be a measurable function in…
Syang Chen
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Elementary proof that monotone functions are differentiable somewhere

It is well-known that every monotone function $f : \mathbb{R} \to \mathbb{R}$ is differentiable almost everywhere (with respect to Lebesgue measure). It is also known if $E$ has measure $0$, then there exists a continuous, monotone function that is…
mrf
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Extending a function by continuity from a dense subset of a space

I am given two spaces $X$ and $Y$, both Hausdorff. I have defined a uniformly continuous function on a dense set $D$ of $X$ that goes to $Y$. So once you have defined a function on a dense subset, if you would like to extend this function to the…
alice
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8 answers

Difference Between Limit Point and Accumulation Point?

I want to clarify the definition of limit point and accumulation point. According to many of my text books they are synonymous that is $x$ is a limit/accumulation point of set $A$ if open ball $B(x, r)$ contains an an element of $A$ distinct from…
mathnoob
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5 answers

Prove that the number of jump discontinuities is countable for any function

I would like to prove that the number of simple jump discontinuities of any function is countable. Can someone point me some material where the proof is or explain the proof here? Thanks.
elaRosca
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