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 …

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 …

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. …

This is also how Weierstrass' famous continuous everywhere, differentiable nowhere function was proven to ...

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 …

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 8, 2017 · continuous spectrum $\sigma_c(T)$: (R2) isn't satisfied, but (R1) and (R3) are satisfied. residual spectrum $\sigma_r(T)$: (R3) isn't satisfied, (R1) is satisfied, (R2) - doesn't …

Aug 19, 2022 · The function ƒ(x) = 1/x (Figure 2.41) is a continuous function because it is continuous at every point of its domain. The point x = 0 is not in the domain of the function ƒ, …

Added @Dimitris's answer prompted me to mention, beyond the fact that the implication on normed spaces indeed is an equivalence, that it's the converse which holds in the wider …