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 a continuous extension? - Mathematics Stack Exchange
To find examples and explanations on the internet at the elementary calculus level, try googling the phrase "continuous extension" (or variations of it, such as "extension by continuity") simultaneously …
What's the difference between continuous and piecewise continuous ...
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
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 …
Prove that $\sqrt {x}$ is continuous on its domain $ [0, \infty).$
As you have it written now, you still have to show $\sqrt {x}$ is continuous on $ [0,a)$, but you are on the right track. As @user40615 alludes to above, showing the function is continuous at each point in the …
Continuous functions that are not uniformly continuous.
Sep 30, 2020 · The proof of the statement relies on showing that continuous functions defined in an interval are uniformly continuous. As i finished doing that part, I started wondering: what are all the …
Proof that the continuous image of a compact set is compact
I know that the image of a continuous function is bounded, but I'm having trouble when it comes to prove this for vectorial functions. If somebody could help me with a step-to-step proof, that would be great.
Can a function have partial derivatives, be continuous but not be ...
Sep 18, 2020 · By differentiability theorem if partial derivatives exist and are continuous in a neighborhood of the point then (i.e. sufficient condition) the function is differentiable at that point.
Is there any intuition for continuous embeddings?
Jan 1, 2026 · Let X, Y X, Y be topological spaces and X ⊂ Y X ⊂ Y. X X is said to be continuously embedded in Y Y if the inclusion map i: X → Y i: X → Y, x ↦ x x ↦ x, is continuous. The definition …
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 …