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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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Convergence of a sequence of functions and their inverses

consider a sequence of continuous and bijective functions $f_n:\mathbb{R}\rightarrow\mathbb{R}$, such that their inverses $f^{-1}_n:\mathbb{R}\rightarrow\mathbb{R}$ are continuous as well. Furthermore let us assume the function…
Braten
  • 1,955
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How much a càdlàg (i.e., right-continuous with left limits) function can jump?

I have changed the title (replaced "well-behaved" by "càdlàg"), since it seems that "a well-behaved function" might be interpreted as "a function of bounded variation" (rather than "a càdlàg function", which I actually meant). Let $f:[0,1] \to {\bf…
Shai Covo
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How prove $g(x)$ is odd function :$g(x)=-g(-x)$

QUestion: let $f(x),g(x)$ is continuous on $R$,and such $$f(x-y)=f(x)f(y)-g(x)g(y)$$ and $f(0)=1$ show that: for any $x\in R$, have $g(x)=-g(-x)$ my try: let $x=y=0$,then $$f(0)-[f(0)]^2=g(0)g(0)\Longrightarrow g(0)=0$$ and let $x=0$, note…
math110
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Limit of Lebesgue integral $\int_F \; \frac{dx}{2 - \sin nx}$

Let $F\subset\mathbb{R}$ be a measurable set of finite Lebesgue measure. Find the limit $$ \lim_{n \to \infty} \int_{F} \frac{dx}{2-\sin nx}.$$ Let $f_{n}(x)=\frac{1}{2-\sin nx}$. This function is bounded and continuous. Thus, it is integrable. I…
user16859
  • 705
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Uniformly bounded derivative implies uniform convergence

Let $f_n$ be a sequence of differentiable functions on $[a, b] \subset \mathbb{R}$. Suppose $\lim_{n \rightarrow \infty} f_n(x) = f(x)$ exists for all $x \in [a, b]$, and the derivatives $|f_n'(x)| < M$ are uniformly bounded over $n$ and…
user09812093
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Is $\cup_{k=1}^\infty (r_k-\frac{1}{k}, r_k+\frac{1}{k}) = \mathbb{R}$?

Let $r_k$ be the rational numbers in $\mathbb{R}$. (1).Is $\cup_{k=1}^\infty (r_k-\frac{1}{k^2}, r_k+\frac{1}{k^2}) = \mathbb{R}$? (2).Is $\cup_{k=1}^\infty (r_k-\frac{1}{k}, r_k+\frac{1}{k}) = \mathbb{R}$? (1).Because $m(\mathbb{R})=+\infty,…
Shine
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When does the integral preserve strict inequalities?

Hi everyone: Suppose that $f(t)$ is a continuous function on $[a,b]$, where we also have $$\alpha
user127474
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Relation between total variation and absolute continuity

Happy Thanksgiving to you all. I have made an attempt to a homework problem, that I'll need someone look over for me. The question is a follows. If $f$ is of bounded variation on $[a,b]$ then $f'(x)$ exists a.e. and$$\int_a^b |f'|~dx\leq…
Nana
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Differentiability of Convolutions

Let $f(x) \in L^p(\mathbb{R})$ and $K \in C^m(\mathbb{R})$. Can I then say that $(f \ast K) (x) = \int_{\mathbb{R}} f(t) K(x-t) dt$ is in $C^m$? I know that this is true if $K$ has compact support, but I was wondering if it is possible to have a…
user1736
  • 8,573
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Prove that a uniformly convergent convergent sequence of $N^\text{th}$ degree polynomials must converge to some $N^\text{th}$ degree polynomial

So here's the question I'm trying to answer: Suppose $p_n(x) = \sum_{k=1}^N a_k^{(n)} x^k$ is a sequence of polynomials such that $p_n \to f$ uniformly over $[0,1]$ for some function $f:[0,1] \to \mathbb{R}$. Prove that $f$ must itself be an…
Ben Grossmann
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Special subset of an Euclidean space

This is a nice problem that I found it somewhere and thought to share it with everyone! Does there exist a subset $S \subset \mathbb{R}^n$ s.t. for every non-zero $t \in \mathbb{R}^n\;, \; S \cap (S+t)$ has precisely one element?
Ehsan M. Kermani
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What does recursive cosine sequence converge to?

I have a sequence defined as follow: $a_0 = 1, a_n=\cos\left(a_{n-1}\right)$. I want to count $\lim_{n\rightarrow\infty} a_n$ - it definitely does have limit by looking at the graph, the first few numbers of the limit are 0.7390851, but I have no…
Karel Bílek
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Prove that $\sqrt{x}$ is continuous on its domain $[0, \infty).$

Prove that the function $\sqrt{x}$ is continuous on its domain $[0,\infty)$. Proof. Since $\sqrt{0} = 0, $ we consider the function $\sqrt{x}$ on $[a,\infty)$ where $a$ is real number and $a \neq 0.$ Let $\delta=2\sqrt{a}\varepsilon.$ Then, $\forall…
eChung00
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Construction of a function which is not the pointwise limit of a sequence of continuous functions

This is somewhat linked to a prior question of mine which was looking to see if a proof of mine regarding the Dirichlet function was correct (it wasn't). I now have an answer to the question which can be answered without using directly using the…
Andrew D
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Negating the Definition of a Convergent Sequence to Find the Definition of a Divergent Sequence

My task is to write a precise mathematical statement that "the sequence $(a_n)$ does not converge to a number $\mathscr l$" So, I have my definition of a convergent sequence: "$\forall\varepsilon>0$ $\exists N\in\Bbb R$ such that $|x_n -\mathscr…
Vladimir Nabokov
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