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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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Prove that a function is smooth if it is smooth in almost all directions

Question So suppose we have a function $f:\mathbb R^2\to \mathbb R$ for which it is given that $x\mapsto f(x,g(x))$ is smooth (i.e., $C^\infty$) for all smooth functions $g:\mathbb R\to\mathbb R$. Can we prove that $f$ is smooth as well? I don't…
Inzinity
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The adherent values of $x_n=cos(n)$ are the interval $[-1,1]$

This question seems really hard, I'm trying to prove that the set of the adherent values of the sequence $x_n=\cos (n)$ is the closed interval $[-1,1]$, i.e., every point of this interval is a limit of a subsequence of $x_n$, and also the limit of…
user42912
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Open balls in $\mathbb{R}^d$ are Jordan Measurable

I'm trying to solve the following question from Terrence Tao's An Introduction to Measure Theory. Show that an open Euclidean ball $B(x, r) := \{y \in \mathbb{R}^d : |y − x| < r\}$ in $\mathbb{R}^d$ is Jordan measurable, with Jordan measure…
user82261
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Condition for $\displaystyle \lim_{x\to 0}f(x)=\lim_{n\to\infty}f\left(\frac{1}{n}\right)$?

Let $f:{\Bbb R}\to{\Bbb R}$. Is there a courterexample for the following equality or is it always true? $$\lim_{x\to 0}f(x)=\lim_{n\to\infty}f\left(\frac{1}{n}\right)$$ What I think is that one might need a non-continuous function since this is…
user9464
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If $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x \to \infty} \frac {f(x)}{x} = L$

I'm trying to solve this question: Let $f:[0,+\infty) \to \mathbb{R}$ be derivable and $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x\to \infty}\frac {f(x)}{x}=L$. I'm trying to solve this question using l'Hôpital rule, but I couldn't use it,…
user42912
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How to prove $f(x) = 4x^{3} + 4x - 6$ has exactly one real root?

How can I show that $f(x) = 4x^{3} + 4x - 6$ has exactly one real root? I think the best way is to show $f'(x) = 12x^2 + 4 > 0$ for all $x \in \mathbb{R}$. Thus, $f'(x)$ has zero real roots. Thus, $f(x)$ has at most one real root. I thought…
user381493
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Lebesgue Dominated Convergence Theorem with Convergence in Measure

I have an issue with the solution to the following problem. I now want to prove that the Lebesgue Dominated Convergence Theorem still works when the condition $\{ f_{n} \}$ converges to $f$ a.e. is replaced by $\{ f_{n} \}$ converges to $f$ in…
BM Yoon
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One-sided Taylor's expansion

Suppose a function $f(t)$ is defined only on $[t_{0},\infty)$. Suppose all "right'' derivatives $f^{(n)}(t)$ exist, that is, $$f^{(1)}(t_{0})=\lim_{\delta\rightarrow 0+}\frac{f(t_{0})-f(t_{0}+\delta)}{\delta}<\infty,$$ and in…
Student of Statistics
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open set is the disjoint union of a countable collection of open intervals

I don't understand that $\{I_x\}_{x\in O}$ is disjoint. For example, $O = \{(1 , 2)\}$. Let $x = 1.5$. Then, $a_x = 1$, and $b_x = 2$. Therefore, $I_x = (1, 2)$. Let $y = 1.6$. Then, similarly, $I_y = (1, 2)$. That is, $I_x$ and $I_y$ are not…
shk910
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Examples of functions where $f'(x)=f(f(x))$ for all $x$

I am looking for examples of functions $f:\mathbb R \to \mathbb R$ where $f'(x)=f(f(x))$ for all $x$. The only example I can find is the trivial one where f is identically 0.
user544680
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proof of the surjectivity of a function that satisfies certain properties

Let $f\colon \mathbb R^2 \to \mathbb R^2$ be a continuous function such that $|f(p)-f(q)|\geq a|p-q|,\quad \forall p,q\in\mathbb R^2$ and $a>0$. Show that $f$ is injective and surjective (therefore has inverse) and that its inverse is…
nom
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Evaluation of the double integral $\int_{[0,1]×[0,1]} \max\{x, y\} dxdy$

Evaluate: $$\int_{[0,1]×[0,1]} \max\{x, y\} dxdy$$ I am totally stuck on it. How can I solve this?
user58430
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Prove that the following set is dense

It is hard for me to show that the set $\{\sqrt{m}-\sqrt{n}; m,n\in \Bbb N\}$ is dense in $\Bbb R$. Please help me.
Aliakbar
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The convergence in Lebesgue differentiation theorem

Let $f$ be a function on $\mathbb R$ such that $f$ is locally integrable. It is well known that from the Lebesgue differentiation theorem we have $$ \frac{1}{h}\int_t^{t+h} u(s)\,ds \to u(t) $$ almost everywhere if $h \to 0$. My question is, can we…
Totoro
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Is it true that $b^n-a^n < (b-a)nb^{n-1}$ when $0 < a< b$?

A Real Analysis textbook says the identity $$b^n-a^n = (b-a)(b^{n-1}+\cdots+a^{n-1})$$ yields the inequality $$b^n-a^n < (b-a)nb^{n-1} \text{ when } 0 < a< b.$$ (Note that $n$ is a positive integer) No matter how I look at it, the inequality seems…
Mr Prof
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