First of all the first link explicitely constructs (in a very strong sense) an explicit bijection R->R^2 so I don't know how why you seem to think otherwise.

It is true that there is no continuous bijection R -> R^2. It is also true that this has nothing to with cardinality, which is only about the existence of bijections (and indeed is defined for arbitrary sets with no topological structure for which continuity makes no sense)

I don't think that it's actually formulating a bijection across the reals - the generator function looks to me to be generating the complete set of rationals.

> Represent each real number x∈[0,1) as a sequence of DIGITS, where each DIGIT is either in {0,…,8} or is of the form 10∗(10k−1)+i with k≥1 and i∈{0,…,8} (i.e., in {90,…,98;990,…,998;9990,…,9998;…}.

He's iterating to infinity, that sure as shit sounds countable to me.

He's associating to each x in [0,1) an infinite sequence of natural numbers (just like decimal representations do), and there's uncountably many such sequences

I may have misread that one, but I still think the reverse mapping only gets as far as within arbitrary bounds.

But he's defining the numbers with a property that makes them countably infinite?

The part that seems problematic is "each real number" the reals exist across bounds, I'm not sure there can be an "each".

I think his iterator is generating infinitesimals of rational and not continuous nature.

Because continuity is not required here. There are non-continuous bijections between R and R^2.

How is continuity not required?

If there are holes in the map does it really cover the whole space?

I think you have misconceptions about what continuity means and I’m not even sure what you mean with holes.

As a simpler example: The map that multiplies each rational number by -1 and keeps irrational numbers as they are is bijective (it’s its own inverse), however it is nowhere continuous.