Can someone explain me these statements?

Question

1) Gases are composed of molecules which collide with one another and solid objects. If density and velocity are taken to be well-defined at infinitely small points, and are assumed to vary continuously from one point to another, the discrete molecular nature of a gas is ignored.

2) The continuity assumption becomes less valid as a gas becomes more rarefied.

In statement 1, what do you mean by discrete molecular nature? I think "assumed to vary CONTINUOUSLY" is in contradiction with "DISCRETE". But, it is the molecules that are discrete and not the molecule's nature or properties.

In statement 2, does the assumption become less valid because "density and velocity" are no longer "well-defined at infinitely small points" ?

These I came along while studying "Aerodynamics" in Wikipedia. The sub-topic being "Continuity".

Answer 1

If you partition a volume of gas to small cubes, and counted the number of molecules in each cube, you could find density to be a more or less continuously varying quantity, but if your cubes are small enough, you would get to a state in which most of the cubes are empty, and a few of them have one molecule in them.

This phenomenon is referred to as the "discrete molecular nature of the gas". i.e. the fact that the gas is composed of empty space, and a few molecules  This is the nature of the gas...

As the gas becomes rarefied, you need larger cubes to be able to ignore the real molecular nature of the gas, and replace it with the continuous quantity description (of densities and average velocities).