The application of Statistical Field Theory, based on the assumption that some physical quantities depend only on long wavelength properties of macroscopic systems, is ubiquitous in many branches of physics. In many cases, however, it is difficult to directly test the validity of the continuum assumption. Here we consider spatially inhomogeneous polymer chains undergoing equilibrium thermal fluctuations and analytically calculate two physical quantities using both a continuum approach (field theory) and its discrete counterpart. First, we calculate a fluctuation-induced adsorption force associated with free-energy variations with the size and strength of an inclusion (formed by adsorbing molecules). Second, inspired by Gaussian network models of protein unfolding, we calculate the spatial distribution of bond strain and energy fluctuations of an inhomogeneous polymer. In both examples, we demonstrate significant discrepancies between the continuum and discrete results, marking the breakdown of the continuum approach. Possible implications, e.g. for fluctuation-induced entropic forces, are discussed.