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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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Show that $\operatorname{int}(A \cap B)= \operatorname{int}(A) \cap \operatorname{int}(B)$

It's kind of a simple proof (I think) but I´m stuck! I have to show that $\operatorname{int} (A \cap B)=\operatorname{int} (A) \cap \operatorname{int}(B)$. (The interior point of the intersection is the intersection of the interior point.) I thought…
HipsterMathematician
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Sufficient and necessary condition for continuously differentiable

I was trying to prove the following statement: Let $U\subset\mathbb{R}^m$ be open and $f:U\to\mathbb{R}^n$. Show that $f$ is continuously differentiable if, and only if, for each $x\in U$, there exists a linear operator…
Yuki
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Monotone increasing function can be expressed as sum of absolutely continuous function and singular function

I'm working on a problem from Royden's Real Analysis: Show that if a function $f$ is monotone increasing on $[a,b]$, then $f$ can be represented as the sum of an absolutely continuous function and a singular function. I understand the general…
bananabanana
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Let $C \subseteq [0,1]$ be uncountable, show there exists $a \in (0,1)$ such that $C \cap [a,1] $ is uncountable

Let $C \subseteq [0,1]$ be uncountable, show there exists $a \in (0,1)$ such that $C \cap [a,1] $ is uncountable From what I know so far if something is countable then it has the same cardinality as $\mathbb{N}$, so to show there exists an $a \in…
oliverjones
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A+B is closed if one of them is compact

Q. Show that if A and B are closed subsets of $R^n$ and one of them is compact then A+B is closed. My doubt: A+B is not necessarily closed given A and B are closed. I need hints to start this question since it involves the concept of compactness.
Foggy
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Prove that $f(x)=0$ for all $x\in\mathbb{R}$

The problem at which I am currently stuck is, Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $f(m+n\sqrt{2})=0$ for all $m,n\in\mathbb{Z}$. Prove that $f(x)=0$ for all $x\in\mathbb{R}$. I have noted that to solve this problem…
user170039
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To find maximum possible value of this integral

If $\int_{0}^{1} f dx=3$ and $\int_{0}^{1} xf dx =2$, then find the maximum value of $$\int_{0}^{1} f^2 dx.$$ What methods would apply to find this maximum value? I am not approaching the methods...
Parvesh Kumar
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Sine and Cosine Expansion Problem

We know: $$\sin{x} = x - \frac{x^3}{3!} + \frac{x^3}{5!}-\dotsb$$ and so on. Also, $$\cos{x} = 1 - \frac{x^2}{2!} +\frac{x^4}{4!}-\dotsb$$ and so on. With the help of these expansions we need to prove that $\sin^2 x+\cos^2 x=1$ . I tried…
User9523
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If $f:I\to \mathbb{R}$ is differentiable and $f'$ is monotone nondecreasing then $f$ is convex.

Let $I \subseteq \Bbb{R}$ be an open interval. If $f:I\to \mathbb{R}$ is differentiable and $f'$ is monotone nondecreasing then $f$ is convex. I would like a hint to solve this exercise. I applied the mean value theorem but I failed to solve…
roger
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$ f: \mathbb{R}^n \to \mathbb{R}^m $ preserving distances

Let $ f: \mathbb{R}^n \to \mathbb{R}^m $ be a function, that preserves distances. Prove that there exist a linear transformation $T$, and a vector $\mathbf{j} \in \mathbb{R}^m $ such that $ f(\mathbf{x}) = T\mathbf{x}+\mathbf{j}$ for every…
Pilot
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Limit Summation interchanging

Is there a theorem which says when we can interchange the limit and sum as follow: $$\lim_{x\to \infty} \sum_{n=1}^{\infty}f(x,n)= \sum_{n=1}^{\infty}\lim_{x\to \infty}f(x,n)$$ Note: In my case the sum $\sum_{n=1}^{\infty}f(x,n)$ is finite at each…
Casey
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$f(x)$ is continuously differentiable on $[0,1]$, prove that $ (\int_0^1 f(x)dx)^2 \leq \frac {1} {12} \int_0^1 (f'(x))^2 dx $

$f(x)$ is a real function continuously differentiable on $[0,1]$. $f(0) = f(1) = 0$ Prove that $$ \left(\int_0^1 f(x)\mathrm{dx}\right)^2 \leq \frac {1} {12} \int_0^1 (f'(x))^2 \mathrm{dx} $$
leafpile
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Is a single point in euclidean space open, closed, neither or both?

In a euclidean space $\mathbb{R}^k$, is the set consisting of a single point open, closed, neither, or both? I would say that a set $E$ consisting of a single point $p$ doesn't have any limit points, so $E$ contains all of its limit points and is…
aparkerlue
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Uniform Convergence of an Exponential Sequence of Functions

How can I show that for every $\epsilon>0$, there exists an $N\in\mathbb{N}$ such that $$\left|f_n(x)-f(x)\right|=\left|\left(\frac{x}{n}+1\right)^n-e^x\right|<\epsilon$$ whenever $n\geq N$ and $x\in\left[-A,A\right]$? By the way,…
wjmolina
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Find lim sup lim inf

How do I find $\lim \sup\text{ or } \lim \inf$ of $ \sin (\frac{n\pi}{5})$ ? I know the $\sin$ function normally oscillates between $-1$ and $1$ but that obviously is not the answer for $\lim \inf$ and $\lim \sup$.
user215327
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