Interesting question on Lagrangian specification

J Hill

Okay, so I've recently been reading through C. Pozrikdis' Introduction to Theoretical and Computational Fluid Dynamics, and came across an interesting exercise: "Discuss whether it is possible to label all point particles within a finite three-dimensional parcel using a single scalar variable, or even two scalar variables."

Now, mathematically speaking, it should be possible-- the set that contains the volume of a fluid should have an uncountably infinite number of points, but its cardinality should be the same as, say, the unit interval. Therefore, it should be possible to create a mapping from the scalar quantity to the region of space, and should be sufficient (though perhaps impractical) for a Lagrangian coordinates. Does this seem like a valid argument, though?

An argument against this that was posed is that that mapping is not necessarily continuous, but does this condition have to be met?