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 …

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 …

Jan 5, 2016 · As such, $\arctan$ is continuous. If you define $\arctan$ by integrals or power series the result is immediate (the first by the Lipshitz continuity of the indefinite integral and the second from …

Sep 18, 2024 · That the preimage of every basis element is open is sufficient for the inverse function to be continuous is seen on page 103 of Munkres' second ed. of Topology. I hope this makes your life …

Apr 14, 2015 · Which is continuous and one-to-one on $\mathbb R$, but is not differentiable at $0$. This is of course just one example, but in general, any time you "stick" two functions together at a point …

Jan 8, 2017 · The reason one refers to this as "continuous spectrum" Historically had nothing to do with continuity; such spectrum was found to fill a continuum, rather than being discrete.

Then the theorem that says that any continuous function on a compact set is uniformly continuous can be applied. The arguments above are a workaround this.

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 …

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 domain. You can likely …

Jul 14, 2012 · My question is How do I graph a continuous function when I am only given restrictions? For example, Sketch the graph of a function that is continuous everywhere except at x=3, and is …