In this paper we study a new class of functions, which we call (ω, c)-periodic functions.This collection includes periodic, anti-periodic, Bloch and unbounded functions.We prove that the set conformed by these functions is a Banach space with a suitable norm.Furthermore, we show several properties of this class of functions as the convolution invariance.We present some examples and a composition result.As an application, we establish some sufficient conditions for the existence and uniqueness of (ω, c)-periodic mild solutions to a fractional evolution equation.