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 …

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 18, 2024 · With your definition, it's not true: mixture of, say, normal and discrete variables has range $\mathbb R$, but it's CDF isn't continuous. The standard name for random variables that have PDF …

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 ...

Sep 5, 2012 · This might probably be classed as a soft question. But I would be very interested to know the motivation behind the definition of an absolutely continuous function. To state "A real valued …

Then the theorem that says that any continuous function on a compact set is uniformly continuous can be applied. The arguments above are a workaround this.

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 …

Nov 17, 2013 · @user1742188 It follows from Heine-Cantor Theorem, that a continuous function over a compact set (In the case of $\mathbb {R}$, compact sets are closed and bounded) is uniformly …

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.