In continuum mechanics, we disregard the detailed atomic structure of a metal or an alloy and assume that the material is continously distributed throughout the region occupied by the body. This assumption is justified if the dimensions under consideration equal several hundred times the distance between two atoms. Results obtained with this assumption agree with experimental observations provided that these are made over length scales of the order of several micrometers or millimeters. Remember that without a continuum theory of matter, one could not have designed an airplane or an automobile or any other structural component. Everyday experience shows that these structural components function properly and safely.
Even though results computed from the continuum theory agree with experimental observations only when the strain gage (or the pressure gage) measures the quantity over a length (or area) of a few mm (or mm2), in deriving the theory we often use differentiation and integration rules. That is, the laws of continuum mechanics are assumed to hold over infinitesimally small lengths, and also in the limit as the length of the element goes to zero.