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 …

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 …

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. …

A natural class of examples would be paths of Brownian motion. These are continuous but non-differentiable everywhere. You may also be interested in fractal curves such as the Takagi …

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 $[-1,1]$.

$\begingroup$ @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 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 …

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Jun 13, 2015 · Continuous function maps connected set onto connected set. 1. Continuity property for a closed set. 0 ...

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 …