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(this is the definition in topology and is the right definition in some sense.) the definitions you cite of semicontinuities claim. We know that differentiable functions must be continuous, so we define the derivative to only be in terms of continuous functions. Do you know that $\ell^ {\infty}$ is not separable?

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I am quite aware that discrete variables are those.

A continuous function does not always map open sets to open sets, but a continuous function will map compact sets to compact sets.

I believe that the set of all $\\mathbb{r\\to r}$ continuous functions is $\\mathfrak c$, the cardinality of the continuum. If f is differentiable at a point x, then f must also be continuous at x, locally linear (the reverse does not hold). However, i read in the book "metric spaces" A function is continuous if the preimage of every open set is an open set.

@boink every open set in the product topology is a union of sets of the form $u\times v$. Both discrete and continuous variables generally do have changing values—and a discrete variable can vary continuously with time. I was trying to formalize some things about string motion in physics so i could answer more general questions about it and then i got to a point as to see the limit written below. But then, the fact that differentiable functions are.