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 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 …
real analysis - Are Continuous Functions Always Differentiable ...
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 …
Is derivative always continuous? - Mathematics Stack Exchange
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 to points on a ...
The graph of a continuous function is a topological manifold
Sep 18, 2024 · The graph of a continuous function is a topological manifold Ask Question Asked 1 year, 6 months ago Modified 1 year, 6 months ago
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 …
The definition of continuous function in topology
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 …
Showing that $\arctan$ is continuous - Mathematics Stack Exchange
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 …
real analysis - Prove that every convex function is continuous ...
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 …
Closure of continuous image of closure - Mathematics Stack Exchange
Nov 14, 2012 · Closure of continuous image of closure Ask Question Asked 13 years, 4 months ago Modified 13 years, 4 months ago