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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 …

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 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 …

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 …

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their …

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 …

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 …

The following are theorems, which you should have seen proved, and should perhaps prove yourself: Constant functions are continuous everywhere.

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, …

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 …