Fluids are characterized by the fact that intramolecular distances are much larger than the size of the molecules. In such a discontinuous medium, the terms velocity, acceleration, density and pressure etc. at a point have no meaning. For instance, density is equal to zero at point, if that point does not coincide with a molecule and would be very large if it does coincide with a molecule. Similarly velocity is zero for the first case and equal to the velocity of the molecule in the second case.
In the mathematical description of fluid flow, it is therefore necessary to assume that the flow quantities such as velocity and pressure vary continuously from one point to another. Once we make the assumption that the fluid itself is continuous in its properties, we may describe these properties with continuous functions of space and time and may apply differential equations in the analysis of processes. Equations derived on the basis of this assumption have withstood the test of time and the treatment of a fluid medium as a continuum has firmly been established.