There are other ways a function can be a continuous extension, but probably the most basic way (and likely about the only way you'll see in elementary calculus) is that you have a function …

May 10, 2019 · This fact is useful to resolve this natural question: Let $\{X_i\}_{i=1}^{\infty}$ be i.i.d. random variables uniform over $[-1,1]$.

Why are all differentiable functions continuous, but not all continuous functions differentiable? For the same reason that all cats are animals, but not all animals are cats. Differentiability also …

Jan 27, 2014 · I understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions. …

Oct 22, 2017 · Continuous Models : Probabilistic models with continuous sample spaces differ from their discrete counterparts in that the probabilities of the single-element events may not …

Oct 15, 2016 · A piecewise continuous function doesn't have to be continuous at finitely many points in a finite interval, so long as you can split the function into subintervals such that each …

Aug 4, 2022 · Isn't this violating the definition of continuous stochastic process or is it that I have to keep $\omega$ constant throught out the process ? Also, is $\omega$ in the definition of …

Aug 1, 2017 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for …

$\begingroup$ @user135626: What I wrote is correct. You are misreading it. I'm not saying the derivative is zero, I'm saying that if the derivative exists, the numerator of the difference …

Oct 8, 2021 · The regularity of Brownian paths is expressed quite precisely by it being $\alpha$-Holder continuous for all $\alpha< \frac 12$ but not $\alpha$-Holder continuous for $\alpha …