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 …

If we restrict ourselves to the case of functions which are continuous on the compact interval $[0,1]$, this is in the sense of (classical) Wiener measure, but is likely well beyond the scope …

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. …

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 …

Sep 24, 2015 · Many authors reserve the word discontinuous to be used only for points where the function is defined (and likewise for continuous, but that's obvious). In that case there is a …

Is there an alternative proof of the fact that a real-valued convex function defined on an open interval of the reals is continuous? Since in general convex functions are not continuous nor …

Jul 21, 2020 · Is the derivative of a differentiable function always continuous? My intuition goes like this: If we imagine derivative as function which describes slopes of (special) tangent lines …

Dec 17, 2020 · What these answers miss a little bit I think is the why of what's going on here. A mistake people make I think is assuming that continuous and 'smooth' (in a loose sense here …