  We investigate the emergence of the continuum elastic limit from the
atomistic description of matter at zero temperature considering how locally
defined elastic quantities depend on the coarse graining length scale. Results
obtained numerically investigating different model systems are rationalized in
a unifying picture according to which the continuum elastic limit emerges
through a process determined by two system properties, the degree of disorder,
and a length scale associated to the transverse low-frequency vibrational
modes. The degree of disorder controls the emergence of long-range local shear
stress and shear strain correlations, while the transverse length scale
influences the amplitude of the fluctuations of the local elastic constants.
