If the size of a system is very much larger than the size of the constituents, and more importantly if the system contains very many constituents, then a continuum approximation is usually justified. An example is the thermodynamic limit: let the the number of atoms N and the volume V of a system both become arbitrarily large such that the ratio N/V (the number density) remains constant. Even for very rarefied gases such as in outer space, if a volume large enough to contain very many gas molecules is considered, the gas can often be approximated as a continuum. As an example, sound waves can travel through any gas, however rarefied, if their wavelength greatly exceeds the average distance between the gas molecules. The energy carried by such sound waves would be minuscule as the gas density is minuscule, but they nevertheless can still exist. A continuum approximation is justified in this case because the system ( a sound wavelength) contains very many gas molecules.

Dear Jack: Thank for your clear answer. Thus the continuum appears as an approximation allowing an easy mathematical description of physic phenomena. But when we suppose that there is a wave associated to the electron, that is a continuum, we do a hypothesis without support.

The wave function in quantum mechanics assumes that space and time are continuous. This is certainly at the very least an excellent approximation well above Planck dimensions. Even as Planck dimensions are approached, the nature of space and time seems more like to change gradually, rather than a sudden, sharp, onset of discreteness or atomicity as with the atomicity of a chemical element, e.g., iron.

The difficulty for this approach is that the interferences at very low flux of photons are not really explained. I propose a possible explanation in The wave and the quantum state. In the conclusion I write:

Let us consider the hypothesis of Louis de Broglie of a wave associated with the electron which has led to the discovery of the wave equations. It leads to extend the notion of continuous to the scale of the interactions between the constituents of the atom. Thus the continuous appears to us always intimately linked to the discontinuous. But it is necessary to keep in mind that the point is a mathematical concept that has no volume and hence no physical reality. The continuous subsequently appears a convenience but one as to keep in mind that material reality remains the discontinuous. It was in noting that light came to us from atoms, therefore emitted from a finite set of atoms, that Einstein was brought to propose the hypothesis of the photon. Come back then to the symmetry between the motion of a mobile and the propagation of a wave which led de Broglie to associate a wave to the electron; It is also possible to associate a flow of material particles in the wave, for example of light, thus joining the hypothesis of the photon of Einstein. However if the number of photons becomes small, then the support of the continuum which is the wave can no longer interpret interference; one has then to consider the stocking of the photons where the interferences are observed (see: Interferences and Periodicity).

Dear Xavier, the discrete distribution of particles energy in structural complexes, such as molecules or clusters, depends on the energy step ratio to the averaged thermal energy.

In molecules this ratio is large, but in clusters it is small. For example, the Hydrogen ionization energy is 13.6 eV, but the averaged thermal energy at room conditions is only 1/40-th of eV. So, in atomic physics at room or comparable temperatures the energy distribution is discrete. But inside stars this distribution may be considered as continuous.

The energy steps in clusters due to high mass of particles are lower than 1 meV. So at ordinary temperatures the energy distribution in clusters is continuous. But at temperatures near 1 K the discrete energy distribution in clusters should demonstrate its existence.

Dear Boris: Thank for your comment. For the low energy you agree with the discrete repartition of the energy.  Now for the stars you write “But inside stars this distribution may be considered as continuous”, OK may be considered, but not continuous in reality.

The important point is to realize that the notion of continuous space is an approximation not the reality. In this view it is difficult to support the notion of wave.

Dear Xavier, we can see discrete lines in the stars' radiation. But this radiation goes from the external part of photosphere with temperature around 6000 K. But inside the atmosphere of a star the temperature goes to millions K. This temperature smashes discrete lines and the energy distribution may be considered as continuous.

Dear Boris: First I am not a specialist of astronomy, but can you give few words to explain the millions of degrees K, I am unable to understand how it is possible to suppose a continuous space from the large value of Temperature. There is no comparison between the infinity of the countable and that of the continuous. Sorry for the delay to answer but my purpose is in fact to shed light on the absence of similitude between the continuous and the countable. Regard Xavier.

Dear Xavier, you know that inside stars there are thermonuclear reactions with temperature of particles measured in millions K. But due to radiation from the outer atmosphere the temperature in these outer layers falls to thousands of K. Yellow color of Sun corresponds to 6 thousand K. Blue stars have larger outer temperature, red ones - lower. But inside of all of them there is a huge temperature, measured in millions K.

If we say about the Space, far from stars there is a low temperature around 3K. At this temperature atoms, molecules, dust have discrete structure of their energetic levels.

Dear Boris: Thank for your comment. Yes I know that inside stars there are thermonuclear reactions with temperature of particles measured in millions K, as a result the particles are countable and not continuous, QED.