In this talk, we will give a brief introduction to continuous representation theory, which arises recently from persistence theory in TDA. In my recent joint work with J.Rock, we construct a continuous analogue of Nakayama algebras using representations of the real line and of the circle. We study the equivalences between these categories of continuous Nakayama representations and show the classification of the categories of continuous Nakayama representations is related with low-dimensional dynamical systems.