These two hypotheses do not automatically amount to a continuum treatment. What we want from a continuum theory - as opposed to a microscopic theory - is to describe an object by continuous/smooth functions, which could be made to obey a set of partial differential equations.

E.g., the microscopic density of a liquid is substantially non-zero only at the points where the atoms/molecules are located, and nearly zero between them. For many purposes such a description is unnecessary, and we can do well with considering density over some microscopically small volume, containing millions of atoms - as we do in everyday life.

Yet, this microscopically small volume should be sufficiently small, as to make the description meaningful, i.e., for the quantities to change on the scale of interest.

Finally, to add a few terms:

• Coarse graining is the terms used to describe mathematically transition from microscopic to macroscopic description.
• Some of the disciplines concerned are: hydrodynamics, elasticity theory, macroscopic electrodynamics, etc.

These two hypotheses do not automatically amount to a continuum treatment. What we want from a continuum theory - as opposed to a microscopic theory - is to describe an object by continuous/smooth functions, which could be made to obey a set of partial differential equations.

E.g., the microscopic density of a liquid is substantially non-zero only at the points where the atoms/molecules are located, and nearly zero between them. For many purposes such a description is unnecessary, and we can do well with considering density over some microscopically small volume, containing millions of atoms - as we do in everyday life.

Yet, this microscopically small volume should be sufficiently small, as to make the description meaningful, i.e., for the quantities to change on the scale of interest.

Finally, to add a few terms:

• Coarse graining is the terms used to describe mathematically transition from microscopic to macroscopic description.
• Some of the disciplines concerned are: hydrodynamics=fluid mechanics, elasticity theory, macroscopic electrodynamics, etc.