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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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Determine the limit $\lim_{k \to \infty} \int_{0}^1 x^{- \frac{1}{2}}\cos(x^k)e^{-\frac{x^2}{k}} dx.$

Determine the limit $$\lim_{k \to \infty} \int_{0}^1 x^{- \frac{1}{2}}\cos(x^k)e^{-\frac{x^2}{k}} dx.$$ I suppose the dominated convergence theorem would be in place here? If I denote $f_k(x)= x^{- \frac{1}{2}}\cos(x^k)e^{-\frac{x^2}{k}}$, then…
Jiming Le
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Analysis Day 1: Few Questions about Cauchy sequences

Sorry for asking these questions; in the textbook I'm using, there are instances where I just don't get how what the author said in paragraph B follows from paragraph A. Why do we need to use Cauchy sequences to define the real numbers? Can't we…
beginner
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problem about continuity and limits

Let $f\colon\mathbb R \to \mathbb R$ be a continuous function. Suppose that $\lim_{x \to +\infty} f(x) = \lim_{x \to -\infty} f(x) = +\infty$. Prove that $f$ has a minimum, i.e., $\exists x_0 \in \mathbb R: \forall x \in \mathbb R f(x) \geq…
Walter r
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Find function $f$ such that $f(1) = 1$ and $f'(x) = \frac{1}{x^2 + \left\{ f(x) \right\}^2}.$

This was actually from my recent exam questioning : Prove $$f(\sqrt3) < 1 + \frac{\pi}{12}$$ for function $f$ such that $f \in C(1, \infty), \:f(1) = 1$ and $$f'(x) = \frac{1}{x^2 + \left\{ f(x) \right\}^2}.$$ The solution is : Since $f'(x) > 0$,…
Vue
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why $g_n$ is measurable in the proof of Fatou's Lemma

Fatou Lemma: Suppose $\{f_n\}$ is a sequence of measurable functions with $f_n \geq 0$. If $\lim_{n\rightarrow\infty}f_n(x)=f(x)$ for a.e. $x$, then $$\int f \leq \liminf_{n\rightarrow\infty}\int f_n$$ Proof: Suppose $0\leq g \leq f$, where $g$ is…
Laura
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Show that derivative less than 1 implies contraction.

I am told that f has a continuous derivative and that $a \leq f(x) \leq b$ and $|f'(x)| < 1 \ \forall x \in [a,b]$ and I have to show that $f$ is a contraction. Now if I take any $x,y \in [a,b]$, the Mean-Value Theorem says that $\exists c \in…
Wooster
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Prove $(\sin{x})^{'} = \cos{x}$

Using $$ \lim_{x \to 0} \frac{\sin{x}}{x} = 1 $$ and summation theorem for sine, prove: $(\sin{x})^{'} = \cos{x}$ So I wrote: $$ \lim_{h \to 0} \frac{\sin{(x+h)} - \sin{x}}{h} = \lim_{h \to 0} \frac{2\sin{(\frac{h}{2})}\cos{(x + \frac{h}{2})}}{h}…
NightEye
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Is proof that this metric is not induced by a norm correct?

Let $d: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, (x, y) \mapsto |e^y - e^x|$ be a metric on $\mathbb{R}$. I want to show that this is not induced by a norm. Claim: d is not induced by a norm. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be…
tor
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Find the minimum ...

Given a positive integer $n$, find the minimum value of $$\frac{x_1^3+x_2^3+...+x_n^3}{x_1+x_2+...+x_n}$$ subject to the condition that $x_1,x_2,...,x_n$ be distinct positive integers I tried and this is what happened Order the numbers $x_1 < x_2 <…
Dmitry
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Differentiable injective function has non-zero derivative?

If $f$ is a real valued injective differentiable function from $(a,b)$, then $f’\neq 0$ for all $x\in (a,b).$ I know that $f’(x)=0$ does not imply local constant. Say for example $x^2$. But I am wondering if the above statement is correct?
MathforFun
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Problem from Zorich's book volume 1

Show that if a function $f$ is defined and differentiable on an open interval $I$ and $[a,b]\subset I$, then a) the function $f'(x)$ (even if it is not continuous!) assumes on $[a,b]$ all the values between $f'(a)$ and $f'(b)$; b) if $f''(x)$ also…
RFZ
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Real Mathematical Analysis Prelim problem 4.60

I am stuck on the following problem from Pugh's book: Does there exist a continuous function $$f: [0,1] \rightarrow \mathbb{R}$$ such that $$\int_0^1 xf(x)\,dx = 1$$ and $$\int_0^1 x^n f(x)\,dx = 0$$ for $$n = 0,2,3,\ldots$$ The progress I made so…
Madhav
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Oscillation less than epsilon gives property near x

I'm working through a Real Analysis textbook and came across this problem that I am having trouble making rigorous Let $f:[a,b]\rightarrow \mathbb{R}$ be bounded, $\epsilon>0$ and $\omega_f(x) < \epsilon$ for all $x \in [a, b]$. Show for each $x_0…
Alexander
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A convexity inequality

The function $\mathbb{R}^n\to \mathbb{R}$, $x\mapsto |x|^p$ (where $p>1$) is convex and thus the inequality $$|y|^p-|x|^p\ge p(y-x)\cdot x |x|^{p-2}$$ is valid. In some lecture notes of Peter Lindqvist, it is remarked that this inequality can be…
Florian
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How to know a function is integrable or not?

Let say $$ h(x) = \begin{cases}x^2,& x \in \mathbb {Q}\\-x^2,& x \notin \mathbb{Q}\end{cases} $$ Is there a difference between riemann integrable and integrable? And can I just say contiuous function is integrable and hence the above…
Timothy Leung
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