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 …

The continuous extension of f(x) f (x) at x = c x = c makes the function continuous at that point. Can you elaborate some more? I wasn't able to find very much on "continuous extension" throughout the web. …

May 10, 2019 · Of course, the CDF of the always-zero random variable $0$ is the right-continuous unit step function, which differs from the above function only at the point of discontinuity at $x=0$.

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

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 …

Jun 11, 2025 · 6 "Every metric is continuous" means that a metric d d on a space X X is a continuous function in the topology on the product X × X X × X determined by d d.

In general, is a bounded linear operator necessarily continuous (I guess the answer is no, but what would be a counter example?) Are things in Banach spaces always continuous?

As you have it written now, you still have to show x−−√ x is continuous on [0, a) [0, a), but you are on the right track. As @user40615 alludes to above, showing the function is continuous at each point in the …

Nov 17, 2024 · 0 You would need lower hemicontinuity, see, e.g., selection theorems, to get the existence of a continuous selection. If your upper hemicontinuous map ϕ ϕ is a singleton at x0 x 0, …

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 …