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

29 votes
Accepted

How did Bernoulli prove L'Hôpital's rule?

L'Hôpital's rule was first published in Analyse des Infiniment Petits. According to The Historical Development of The Calculus by Edwards (p. 269), L'Hospital's argument, which is stated verbally w …
Martin Sleziak's user avatar
9 votes
Accepted

Quantitative bounds for multivariate central limit theorem

There is a bunch of such statements which can be obtained by Stein's method. You might be interested in the paper "On the Rate of Convergence in the Multivariate CLT" by Gotze, which is specifically …
David Roberts's user avatar
  • 35.5k
13 votes
Accepted

Points of continuity of Baire class one functions

This is impossible. Baire proved that if a function defined on $\mathbb R$ is of Baire class 1, then it is continuous everywhere except, possibly, for a meagre set. And by another Baire's theorem a co …
AMal's user avatar
  • 3
16 votes

Does pointwise convergence imply uniform convergence on a large subset?

To add a bit to Jonas Meyer's answer. Sierpiński's result was first published in C.R. Soc. Sc. Varsovie 1928, p. 84-87. It was reproduced in his monograph "Hypothèse du continu" (Lwów, 1934, p. 52): …
Andrey Rekalo's user avatar
3 votes
Accepted

Cartesian product of test function spaces

This is true. By a partition of unity, the proof can be reduced to the case when the test functions have their supports in a unit cube and the result follows from a more or or less straightforward …
Andrey Rekalo's user avatar
14 votes
Accepted

approximately linear functions

Let $E$ and $E'$ be Banach spaces. Mappings $f:E\to E'$, which satisfy the inequality $$\|f(x + y) − f(x) − f(y)\| \leq\epsilon$$ for all $x, y \in E$, are called $\epsilon$-additive (or approximate …
Andrey Rekalo's user avatar
1 vote

Looking for an interesting problem/riddle involving triple integrals.

Feynman's formulae for products. $$\iiint\limits_{[0,1]^3}\frac{x_1^2 x_2}{[(a_4-a_3)x_3 x_2x_1+(a_3-a_2)x_2 x_1+ (a_2-a_1)x_1+a_1]^4}dx_1dx_2dx_3=$$ $$=\int\limits_{0}^{1}dx_1\int\limits_{0}^{1-x_1}d …
Andrey Rekalo's user avatar
7 votes

Looking for an interesting problem/riddle involving triple integrals.

This is one of my all time favourites (quoted from Problems and Theorems in Analysis by Polya and Szego). The 3D domain $\mathcal D$ is defined by the inequalities $$-1\leq x,y,z\leq 1,\quad -\ …
Andrey Rekalo's user avatar