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 R but not uniformly …

12 Following is the formula to calculate continuous compounding A = P e^(RT) Continuous Compound Interest Formula where, P = principal amount (initial investment) r = annual interest …

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?

May 6, 2016 · That's right. A constant function is continuous, but for most topologies does not map an open set to an open set. For a familiar somewhat different example, the image of $ …

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 …

Nov 17, 2013 · @user1742188 It follows from Heine-Cantor Theorem, that a continuous function over a compact set (In the case of $\mathbb {R}$, compact sets are closed and bounded) is …

Nov 9, 2024 · I learned a theorem that if $f$ is continuous and bijective then $f^ {-1}$ is continuous. I went online to search for a proof and saw a really long proof in this link.

I was reading through Munkres' Topology and in the section on Continuous Functions, these three statements came up: If a function is continuous, open, and bijective, it is a homeomorphism. If a