# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

18,356 questions
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### What functions can be made continuous by “mixing up their domain”?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ so that $f\circ \phi$ is continuous. So one could say a potentially ...
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### On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
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### Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$

Is there a way to assess the convergence of the following series? $$\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$$ From numerical estimations it seems to be convergent but I don't know how to prove it.
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### Constructing an infinite chain of subfields of 'hyper' algebraic numbers?

This has now been cross posted to MO. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn ...
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### Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
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### A subadditive bijection on the positive reals

Question. Does there exist a subadditive bijection $f$ of the positive reals $(0,\infty)$ such that $$\liminf_{x\to 0^+}f(x)=0 \,\,\,\text{ and }\,\,\,\limsup_{x\to 0^+}f(x)=1\,?$$ Ps. I guess ...
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### Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

Update 24 Nov 2015: It is solved. Please refer to this arXiv paper. For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids ...
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If $V$ is a linear subspace of $L^\infty[0,1]$ with $\|f\|_\infty \leq c\|f\|_2$ for all $f\in V$, then $V$ is finite dimensional. The proof is an explicit calculation: Since $L^\infty[0,1] \subset L^... 0answers 532 views ### Bounding a polynomial from below Let$\sigma >0$be fixed. For even$k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \... 0answers 128 views ### How much can we rearrange a series? There's a well-known result that if$\sum a_n$is conditionally convergent, then for any real$c$there exists a permutation$\pi:\mathbb{N} \to \mathbb{N}$such that$\sum a_{\pi(n)} = c$. A ... 0answers 99 views ### A question about Existence of a Continuous function. Let$f$be a continuous function on the interval$[1,2]$. It follows from Stone-Weierstrass theorem that if$\displaystyle \int_1^2x^nf(x)dx=0$for integers$n=0,1,2,....$, then$f$must be ... 0answers 259 views ### Norms and pointwise convergence It is known that There is no norm$\|.\|$on the space$E$of continuous real-valued functions on an interval, say$[0,1]$such that$f_n \to f$for$\|.\|$if and only if$f_n$converges pointwise ... 0answers 680 views ### Lagrange multipliers in Calculus of Variations I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ... 0answers 103 views ###$\frac{1}{p}+\frac{1}{q}=1$vs$\sum_{n=0}^\infty \frac{1}{p^n}=q$It just occurred to me that conjugate exponents, i.e.$p,q\in(1,+\infty)$such that $$\frac{1}{p}+\frac{1}{q} =1$$ also satisfy the relations:$\sum_{n=0}^\infty \frac{1}{p^n}=q;\sum_{n=0}^\infty \...
For any function $f$ denote by $Z(f)$ and $Z_o(f)$ the cardinalities of $f^{-1}(0)\cap[0,1]$ and $f^{-1}(0)\cap(0,1)$, respectively. Let $H=\{f\in C^\infty(\mathbb{R}): \text{supp}(f) = [0,1]\}$ From ...