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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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the Riemann integrability of inverse function

If $f \colon [a,b] \rightarrow [c,d]$ is a bijection, $f\in \mathcal{R}$ and $f^{-1}$ exists, then prove or disprove that $f^{-1} \in \mathcal{R} [c,d]$. Remark: I tried to use integration by parts to find $\int_{c}^{d} f^{-1}$ and to prove that was…
Junyu
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Triangle inequality for infinite number of terms

We can prove that for any $n\in \mathbb{N}$ we have triangle inequality: $$|x_1+x_2+\cdots+x_n|\leqslant |x_1|+|x_2|+\cdots+|x_n|.$$ How to prove it for series i.e. $$\left|\sum \limits_{n=1}^{\infty}a_n\right|\leqslant \sum…
Raheem Najib
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Dedekind Cuts versus Cauchy Sequences

Are there any advantages or disadvantages in defining a real number in the following ways: Definition 1 A real number is an object of the form $\lim\limits_{n \to \infty} a_n$ where $(a_n)_{n}^{\infty}$ is a cauchy sequence of rational numbers.…
NebulousReveal
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Differentiability of Norms

I am looking at for which point(s) in $\mathbb{R}^n$ and $p\geq 1$ such that the map $\mathbb{R}^n\rightarrow\mathbb{R}$ given by $x\mapsto ||x||_p=(|x_1|^p+\cdots+|x_n|^p)^{\frac{1}{p}}$ is differentiable. Intuitively, it is not hard to "guess"…
iloveinna
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Subset of $\mathbb{Q}$

Let $S= \{x_0,\dots,x_n\}$ be a finite subset of $[0,1]$ , $x_0=0$ and $x_1=1$ such that every distance between pair of elements of $S$ occurs at least twice, except for the distance $1$, then we are to show that $S$ is a subset of $\mathbb{Q}$.
Myshkin
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Does every uncountable subset of $\mathbb{R}$ have an uncountable closed subset?

Let $E\subseteq \mathbb{R}^1$ be an uncountable set. Can we obtain some subset $F\subseteq E$ which is closed and uncountable? Basically, I want to construct some set containing only irrational numbers which is also uncountable and closed, in a…
user119882
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Can we construct a function that has uncountable many jump discontinuities?

I know that Dirichlet function has uncountable many discontinuities. I think they are removable, because the discontinuities can be removed by redefining the function values of the rational numbers as 0. So Dirichlet function is a function that has…
Tony
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Writing $f(x,y)$ as $\Phi(g(x) + h(y))$

Could you prove or disprove the following statement? Let $f\colon[0,1]^2\rightarrow \mathbb R$ be a continuous function. Then there are continuous functions $g,\ h\colon [0,1]\rightarrow \mathbb R$ and $\Phi\colon \mathbb R \to…
AgCl
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Any explicit examples of irrationals in the Cantor set?

Since the Cantor set is uncountable, it must contain irrationals. I am aware that they can't be normal, so the irrationals in the Cantor set are transcendental. Are there any explicit constructions of such numbers, or can we only indirectly show…
cantorset
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Is irrational times rational always irrational?

Is an irrational number times a rational number always irrational? If the rational number is zero, then the result will be rational. So can we conclude that in general, we can't decide, and it depends on the rational number?
Firas Abd El Gani
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Derivative of a function is odd prove the function is even.

$f:\mathbb{R} \rightarrow \mathbb{R}$ is such that $f'(x)$ exists $\forall x.$ And $f'(-x)=-f'(x)$ I would like to show $f(-x)=f(x)$ In other words a function with odd derivative is even. If I could apply the fundamental theorem of…
user9352
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Let ($x_n$) be a monotone sequence and contain a convergent subsequence. Prove that ($x_n$) is convergent.

Let ($x_n$) be a monotone sequence and contain a convergent subsequence. Prove that ($x_n$) is convergent. I know that by the Bolzano-Weierstrass Theorem, every bounded sequence has a convergent subsequence. But I need some hints as to how to prove…
Jason
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Real sequences which sum to 0, multiply to 1.

Does there exist two sequences $(x_n)_n$, $(y_n)_n$ of real numbers such that $\lim_n x_n-y_n\neq 0$ (may not exist), but $\lim_n x_n+y_n=0$ and $\lim_n x_ny_n=1$? Notice that it cannot be the case that both sequences are convergent, or even…
Iian Smythe
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Countable sum of measures is a measure

Prove that if $\mu_1, \mu_2, \dots$ are measures on a measurable space and $a_1, a_2, \dots \in [0,\infty)$, then $\sum_{n=1}^\infty a_n\mu_n$ is also a measure. I need some help justifying the third equality in the final line of my proof. My idea…
dannum
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What properties are used to assert that there is always a number between two given numbers?

What properties of "numbers" are used to assert that for given numbers $a$ and $b$, $a≠b$, there exists a number $x$ such that $a < x < b$ ? In the texts I've read, this seems to be assumed without explanation in discussions that are otherwise quite…
orome
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