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 …

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 …

Jul 22, 2010 · The cardinality is at most that of the continuum because the set of real continuous functions injects into the sequence space $\mathbb R^N$ by mapping each continuous function to its …

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 …

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.

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.

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 ...

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 …

The point is the probability that two continuous random variables assume the same value is $0$ which gives $\alpha = 1/2$. Had they been discrete, then yes one would very much need to account for that …

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 …