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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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When is a uniformly dense family of functions dense in L^p?

Suppose $\mathcal{A}\subset L^p(\mathbb{R})$ is an algebra of functions with the following property: For every compact $K\subset\mathbb{R}$, $\mathcal{A}$ is dense in $\mathcal{C}(K)$ with respect to the uniform norm $\|\cdot\|_{\infty}$, where…
Nick Strehlke
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Prove that $e^n\bmod 1$ is dense in $[0,1]$

I just noticed that I have left unanswered one part of an old multi-part question and so decided to re-ask it separately: Consider the sequence $e^n\bmod 1$, $n\in\Bbb N$. Show that it is dense in $[0,1]$. This apparently does require specific…
Hagen von Eitzen
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Convergence of Multidimensional Infinite Series

I am trying to analyze the convergence of multidimensional infinite sums such as those in the following form: $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{1}{1+\alpha\exp (\beta n) \exp (\gamma m)}$$ where $\alpha,\beta,\gamma\in(0,\infty)$. I'm…
Chris M
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Can a function $f:\mathbb{R} \rightarrow \mathbb{R}$ be "infinitely steep" on a set with non-zero Lebesgue outer measure?

By being "infinitely steep" on a set I mean that for each point $x$ in the set we have $\sup\limits_{\delta>0}\text{ }\inf\limits_{y\in(x-\delta,\text{ }x+\delta)\backslash\{x\}}\frac{f(y)-f(x)}{y-x}=\infty$. As a remark, the Cantor function is…
Jonathan Hole
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Prove that $f'(x) \to 0$ as $x \to \infty$ when $f'+f''$ is bounded abve.

I am given $f \in C^2([0,\infty))$ and $\lim_{x \to \infty}f(x) = L \in \mathbb{R}$. I am also given there is a real number $M$ such that $f'(x) + f''(x) < M$ for all $x \in [0,\infty)$. Prove that $\lim_{x \to \infty}f'(x) = 0$. My thoughts…
SAS
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A math analysis problem.

Suppose $f(x)$ is a differentiable function on $\mathbb R$ with continuous derivative. For any $x \in \mathbb R$, $f’(x)>f(f(x))$. Prove that for any $x\ge 0$, $f(f(f(x)))\le 0$. I don’t know how to use the continuity of the function’s derivative…
passerby
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Is the set of discontinuity of $f$ countable?

Suppose $f:[0,1]\rightarrow\mathbb{R}$ is a bounded function satisfying: for each $c\in [0,1]$ there exist the limits $\lim_{x\rightarrow c^+}f(x)$ and $\lim_{x\rightarrow c^-}f(x)$. Is true that the set of discontinuity of $f$ is countable?
Tomás
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Increasing function on R that is discontinuous on the rationals

The question (Folland's Real Analysis, 3.5.30) asks to produce an increasing function on $\mathbb{R}$ whose set of discontinuities is the rationals. My train of thought is as follows: Let $f_0$ be the identity. We want to create a jump at every…
Luca S.
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Is a monotone differentiable function continuously differentiable?

If $f:\mathbb{R}^+\to\mathbb{R}$ is monotonic and differentiable, does it follow that $f$ is continuously differentiable? (This question arose from discussion here: problem on continuous and differentiable function)
user7530
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Calculate $\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$

Please help me calculate this: $$\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$$ Here I've tried multiplying by $\sqrt[4]{x+9}+2$ and few other method. Thanks in advance for solution / hints using simple…
Steve
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How to prove sin(nx) has no pointwise convergent subsequence without prior knowledge of Lebesgue's Theory?

In Baby Rudin 7.20 example, the author mentions to prove that the function sequence $$f_n(x):=\sin(nx) \qquad0(\leq x\leq 2\pi)$$has no pointwise convergent subsequence would be troublesome without Lebesgue's Theorem. Is there a proof that doesn't…
Math The Novice
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Equivalence of Rolle's theorem, the mean value theorem, and the least upper bound property?

How to show that Rolle's theorem, the Mean Value Theorem are equivalent to the least upper bound property? I'm thinking of starting like this: Let F be an ordered field that does not satisfy the least upper bound property, and then deduce that F…
Lindsay Duran
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Product of measurable functions is measurable

Here's the question I'm having a go at: "Prove that if $f$ and $g$ are measurable then $fg$ is also measurable (express the product using sums and powers of functions)" I've had a look for the proof that the pointwise sum of measurable functions is…
user416936
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How to solve an equation of the form $f(x)=f(a)$ for a fixed real a.

I got stuck on this question: find all solutions $x$ for $a\in R$: $$\frac{(x^2-x+1)^3}{x^2(x-1)^2}=\frac{(a^2-a+1)^3}{a^2(a-1)^2}$$ I see that if we simplify we get: $$\frac{(x^2-x+1)^3}{x^2(x-1)^2}=\frac{[(x-{\frac 12})^2+{\frac 34}]^3}{[(x-{\frac…
zagortenay333
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Is every positive real number the limit of a sequence of ratios obtained from a double partition of $ \mathbb{N} $?

Let $ \{ A,B \} $ be a partition of $ \mathbb{N} $ into two infinite subsets. For every $ r \in \mathbb{R}_{> 0} $, can we find an increasing sequence $ (a_{n})_{n \in \mathbb{N}} $ in $ A $ and an increasing sequence $ (b_{n})_{n \in \mathbb{N}} $…
Haskell Curry
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