Intereting Posts

sum of square roots
What is an Homomorphism/Isomorphism “Saying”?
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if a complex function $f$ is real-differentiable, then $f$ or $\overline{f}$ are complex-differentiable
Main branches of mathematics
$GF(n,6)$ is prime iff : $GF(n,6) \mid S_n$?
Modified gambler's ruin problem: quit when going bankruptcy or losing $k$ dollars in all
Show that $\mathbb{Q}^+/\mathbb{Z}^+$ cannot be decomposed into the direct sum of cyclic groups.
On the inequality $ \int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$
function a.e. differentiable and it's weak derivative
Let $p$ be an odd prime number. How many $p$-element subsets of $\{1,2,3,4, \ldots, 2p\}$ have sums divisible by $p$?
Is there a difference between allowing only countable unions/intersections, and allowing arbitrary (possibly uncountable) unions/intersections?
Metric triangle inequality $d_2(x,y):= \frac{d(x,y)}{d(x,y)+1}$
Antiderivative of $e^{x^2}$: Correct or fallacy?
Proof that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$

Could you guide me how to prove that any monotone function from $R\rightarrow R$ is Borel measurable?

Should we separate the functions into continuous and non-continuous? How to prove for not continuous points?

Thanks for your help

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- Unifying the treatment of discrete and continuous random variable

Hint: If $f$ is monotone, then, for every real number $x$, the set $f^{-1}((-\infty,x])=\{t\mid f(t)\leqslant x\}$ is either $\varnothing$ or $(-\infty,+\infty)$ or $(-\infty,z)$ or $(-\infty,z]$ or $(z,+\infty)$ or $[z,+\infty)$ for some real number $z$.

To show this, assume for example that $f$ is nondecreasing and that $u$ is in $f^{-1}((-\infty,x])$, then show that, for every $v\leqslant u$, $v$ is also in $f^{-1}((-\infty,x])$.

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