The continuous extension of f(x) f (x) at x = c x = c makes the function continuous at that point. Can you elaborate some more? I wasn't able to find very much on "continuous extension" …

Oct 15, 2016 · A continuous function is a function where the limit exists everywhere, and the function at those points is defined to be the same as the limit. I was looking at the image of a …

Jan 27, 2014 · To understand the difference between continuity and uniform continuity, it is useful to think of a particular example of a function that's continuous on R but not uniformly …

In general, is a bounded linear operator necessarily continuous (I guess the answer is no, but what would be a counter example?) Are things in Banach spaces always continuous?

6 All metric spaces are Hausdorff. Given a continuous bijection between a compact space and a Hausdorff space the map is a homeomorphism. Proof: We show that f f is a closed map. Let K …

A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it. Until today, I thought these were merely two equivalent definitions of the same c...

The authors prove the proposition that every proper convex function defined on a finite-dimensional separated topological linear space is continuous on the interior of its effective …

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 …

For example, 1 x 1 x is continuous everywhere except perhaps at x = 0 x = 0, by point 6, because it is a quotient of a constant function (point 1) and the identity function (point 2).

Aug 19, 2022 · The MIT supplementary course notes you linked to give — and use — the following (non-standard) definition: We say a function is continuous if its domain is an interval, …