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 …

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 …

Oct 26, 2010 · An interesting fact is that most (i.e. a co-meager set of) continuous functions are nowhere differentiable. The proof is a consequence of the Baire Category theorem and can be found (as an …

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 …

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 …

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.

Mar 7, 2013 · 22 I am self-studying general topology, and I am curious about the definition of the continuous function. I know that the definition derives from calculus, but why do we define it like …

May 11, 2015 · It can be shown that for any 2 functions f and g, if f is continuous on R and g is a linear function with nonzero slope, f ∘ g is continuous so for any positive real number a, if exp (x) is …

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 …

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 …