  We determine which translationally invariant matrix product states have a
continuum limit, that is, which can be considered as discretized versions of
states defined in the continuum. To do this, we analyse a fine-graining
renormalization procedure in real space, characterise the set of limiting
states of its flow, and find that it strictly contains the set of continuous
matrix product states. We also analyse which states have a continuum limit
after a finite number of a coarse-graining renormalization steps. We give
several examples of states with and without the different kinds of continuum
limits.
