Hmm. The field lines are what 'attract' electrons, ions and protons around themselves as I understands it. As you say they are described mathematically, but you can also find them in 'reality', as proven by any experiment with iron fillings and a magnet. I'm not sure it's a 'continuous magnetic field' although one might argue so? Maybe it is in a way, same way as you might argue that even though there is 'gravity' everywhere in SpaceTime you can still, in deep space, find patches where that 'gravity' becomes unmeasurable. Or for that sake argue that in any 'uniform motion' gravity disappear.

"Magnetic field--space modified by magnetic forces, so to speak--is one thing, and magnetic field lines are something else again. They are a mathematical description of that field, no more tangible than lines of latitude and longitude which describe the surface of the Earth. One never asks how close THOSE lines are; you can draw any number of them, depends how tightly you are willing to space them.

Magnetic field lines are defined as lines that point everywhere along the magnetic force (in a fluid, a complete analogy is given by flow lines or "streamlines"). They can be described by formulas, in terms of quantities known as Euler potentials or Clebsch functions.

But there also exist intuitive properties: particles threaded by a common field line, tend to share that field line later on as well. Say we have 10 ions numbered 1... 10 sitting on a common spot on the Sun, and therefore sharing there a field line, and destined to come out in the solar wind one day apart. The Sun rotates, so make a drawing with a circle representing the Sun and 9 radial lines coming out about 15 degrees apart. After 10 days, particle #1 is 2.5 inches along the first line, particle #2 2.25 inches on the 2nd one, and so on, down to particle #10 still on the surface: the line conecting the particles is a spiral, so we expect interplanetary field lines to have a spiral shape, and we derived this from intuitive concepts alone (though the same thing can be derived from formulas).

The details of this excercise are described in

http://www.phy6.org/stargaze/Simfproj.htm.

The spacing between field lines is not meaningful (though some engineers speak of "density of magnetic field lines" to describe a quantity commonly known as "flux density.") Suppose you draw two field lines of the Earth, reaching Earth 1 meter or 1 foot apart. Each can have electrons or ions trapped around it. The meaningful question is what is the radius of the circle these electrons or ions describe around their guiding line, and that depends on their energy, and how strong the field is (the circle gets larger in the weak fields far from Earth), but it is generally much more than 1 foot or 1 meter. No problem: densities are so low that such ions or electrons rarely collide, and their orbits can easily overlap. The radius of gyration of auroral electrons can be 100 meters, which is why auroral "curtains"are so thin. On the other hand, solar wind ions entering near the "nose" of the magnetosphere have radii of the order of 500 kilometers, or (say) 350 miles, because the field there is much weaker, and that is therefore the order of the expected thickness of the magnetopause, the boundary between the solar wind and the magnetosphere." From

By David P. Stern