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 …
Continuous group actions - Mathematics Stack Exchange
Dec 18, 2025 · I was recently going through General Topology by N. Bourbaki, and found the following definition of topological groups acting continuously on topological spaces (slightly rephrased) : A …
Difference between continuity and uniform continuity
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 $\mathbb R$ but not uniformly …
Is the projection function in a normed space continuous?
Dec 26, 2025 · Let (X, n) (X, n) be normed space and B B a basis (by basis I mean a set of vector such that every vector in X can be expressed in an essentially unique way as a finite linear combination of …
A short proof that if $f$ is continuous then $f^ {-1}$ continuous
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.
What does it mean that "every metric is continuous"?
Jun 11, 2025 · 6 "Every metric is continuous" means that a metric $d$ on a space $X$ is a continuous function in the topology on the product $X \times X$ determined by $d$.
Why is/isn't the derivative of a differentiable function continuous?
Jun 19, 2016 · This would mean that the derivative of a function is always continuous on the domain of the function, but I have encountered counterexamples. I have probably misinterpreted something; …
real analysis - Midpoint-convexity and continuity implies convexity ...
Apr 1, 2025 · It's more correct to say that he proved Jensen's Inequality (with arbitrary real weights) for functions which are midpoint convex and continuous. Of course, Jensen's Inequality with two …
Prove that $\sqrt {x}$ is continuous on its domain $ [0, \infty).$
As you have it written now, you still have to show $\sqrt {x}$ is continuous on $ [0,a)$, but you are on the right track. As @user40615 alludes to above, showing the function is continuous at each point in the …
Continuous functions do not necessarily map closed sets to closed sets
May 21, 2012 · I found this comment in my lecture notes, and it struck me because up until now I simply assumed that continuous functions map closed sets to closed sets. What are some insightful …