# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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### Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer https://math.stackexchange.com/questions/1444498/is-there-a-categorizaiton-system-for-null-...
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### A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, we have that $A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$? This question is ...
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### Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
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### Increasing derivatives of recursively defined polynomials

Consider recursively defined polynomials $f_0(x)=x$ and $f_{n+1}(x)=f_n(x)−f_n'(x) x (1−x)$. These polynomials have some special properties, for example $f_n(0)=0$, $f_n(1)=1$, and all $n+1$ roots of ...
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### Proving Richardson's theorem for constants

(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...
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### Uniform approximation of separately continuous functions on zero-dimensional spaces

For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
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### About the first decimal of $\sqrt {n!}$

Do we have : $$\sup\{\sqrt {n!} - E(\sqrt {n!}); n\in I\!\!N\}=1?$$ Where $E(\cdot)$ is the integer part function, and $n!=1\times 2...\times n$.
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### A monoid-structure on pairs of interlacing polynomials

Let us call a pair of two real polynomials $(P,Q)$ interlacing if $\deg(P)=\deg(Q)+1$, both polynomials have strictly positive leading coefficients and $P,Q$ have only real roots which interlace ...
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### Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
### On derivatives of polynomials majorized by $\max(1,|x|^d)$
In the course of generalizing the Bernstein-Markov theorem to normed space, Harris came up with the following question. Suppose that $p$ is a real polynomial satisfying $|p(x)| \leq (1+|x|)^d$. How ...