Continuous vs Discrete Variables - Mathematics Stack Exchange
Dec 14, 2025 · Both discrete and continuous variables generally do have changing values—and a discrete variable can vary continuously with time. I am quite aware that discrete variables are those …
What is the formal definition of a continuous function?
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, and it is continuous …
Prove that the function $\sqrt x$ is uniformly continuous on $\ {x\in ...
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 uniformly …
elementary set theory - Cardinality of set of real continuous functions ...
Jul 22, 2010 · The cardinality is at most that of the continuum because the set of real continuous functions injects into the sequence space $\mathbb R^N$ by mapping each continuous function to its …
Continuous function proof by definition - Mathematics Stack Exchange
More frustratingly, the people giving the answers make bigger mistakes or have bigger confusions about continuity than the person asking for continuity: for a detailed explanation on how to show that the …
Absolutely continuous functions - Mathematics Stack Exchange
Sep 5, 2012 · This might probably be classed as a soft question. But I would be very interested to know the motivation behind the definition of an absolutely continuous function. To state "A real valued …
What is the difference between discrete and continuous mathematics?
Some people like discrete mathematics more than continuous mathematics, and others have a mindset suited more towards continuous mathematics - people just have different taste and interests. On the …
calculus - Is a differentiable function always continuous ...
Sep 14, 2014 · 9 Continuous Functions are not Always Differentiable. But can we safely say that if a function f (x) is differentiable within range $ (a,b)$ then it is continuous in the interval $ [a,b]$ . If so , …
What is the intuition for semi-continuous functions?
A function is continuous if the preimage of every open set is an open set. (This is the definition in topology and is the "right" definition in some sense.) The definitions you cite of semicontinuities claim …
general topology - A map is continuous if and only if for every set ...
Nov 18, 2015 · A map is continuous if and only if for every set, the image of closure is contained in the closure of image