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 …

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 …

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.

Oct 15, 2016 · A continuous function is a function where the limit exists everywhere, and the function at those points is defined to be the same as the limit. I was looking at the image of a piecewise continuous

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 …

Here you want to refer to the topology of the latter as a normed space, which does not depend on the norm since they are all equivalent in finite dimension. Then the determinant is a polynomial in the …

Showing that a function is not continuous Ask Question Asked 9 years, 1 month ago Modified 7 years, 5 months ago

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 …

Jun 20, 2018 · We know that differentiable functions must be continuous, so we define the derivative to only be in terms of continuous functions. But then, the fact that differentiable functions are …