Can you find $A \subset \mathbb R^2$ such that $A, \overline{A}, \overset{\circ}{A}, \overset{\circ}{\overline{A}}, \overline{\overset{\circ}{A}}$ are all different?
Can we get even more sets be alternating again closure and interior?
Let $f:[a,b]\to R$ be a bounded function such that $D:=$ the subset of $[a,b]$ such that $f$ is not continuous at any point of $D$. $f$ is Riemann integrable iff $\lambda (D)=0$.
Proof: For $n\in \mathbb N$, let $P_n$ be the partition that divides $[a,b]$ in $2^n$ equal subintervals $I_j, j=1,2,...
I tried to prove it the following way but got stuck:
Take any two sequences $x_n$ and $y_n$ such that $x_n-y_n\to 0$.
Define $E_n:=[x_n, y_n]\cup [y_n, x_n]$.
$|g(x_n)-g(y_n)|=|\int f\chi_{E_n}d\lambda|$
$|f\chi_{E_n}|\le |f|$ and $\int |f|d\lambda <\infty$. So by the Dominated Convergence Theore...
Given an non-commutative associative unital algebra A of characteristic $0$, one can construct a Jordan algebra $A+$ using the same underlying addition vector space.
Notice first that an associative algebra is a Jordan algebra if and only if it is commutative.
To make it commutative we can define...
Say there exist some non-zero distinct residues $a,b$ such that
$$ a^n + b^n \mod p $$
generates all nonzero residues for some $n$.
Does such a pair $a,b$ exist for every odd prime $p > 13$ ? Or for all sufficiently large prime $p$ ?
And if so, how many pairs exist as a function of $p$ ?
I consid...
