Quick
A continuum is a continuous, homogeneous matter with no holes.
Details
In the gas phase, matter is made up of atoms that are spaced very far apart relative to the size of the atom. Yet it may be convenient to disregard the atomic nature of a substance and view it as a continuum.
The continuum idealization allows properties to be treated as point functions and to assume that the properties vary continually in space with no jump discontinuities (properties hold at one point, then don't hold up at another - click here for jump discontinuity from a mathematical standpoint). This idealization is valid as long as the size of the system under consideration is large relative to the space between molecules. This is the case in most fluid mechanics problems.
Consider a container filled with oxygen at atmospheric conditions. The diameter of the oxygen molecule is about 3 x 10-10 m and its mass is 5.3 x 10-26 kg. Also, the mean free path of oxygen at 1 atm pressure and 20°C is 6.3 x 10-8 m (about 200 times its diameter) before it collides with another molecule.
Also, there are about 2.5 x 1016 molecules of oxygen in the tiny volume of 1 mm3 at 1 atm pressure and 20°C. The continuum model (the assumption of a continuous, homogeneous matter with no holes) is applicable as long as the characteristic length of the system (such as its diameter) is much larger than the mean free path of the molecules. When approaching a complete vacuum or at very high elevations, the mean free path may become large (for example, it is about 01 m for atmospheric air at an elevation of 100 km). For such cases the rarefied gas flow theory should be used, and the impact of individual molecules should be considered.