In this paper we study a new class of functions, which we call $(\omega ,c)$ -pseudo periodic functions. This collection includes pseudo periodic, pseudo anti-periodic, pseudo Bloch-periodic, 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 prove the existence and uniqueness of $(\omega ,c)$ -pseudo periodic mild solutions to the first order abstract Cauchy problem on the real line. Also, we establish some sufficient conditions for the existence of positive $(\omega ,c)$ -pseudo periodic solutions to the Lasota–Wazewska equation with unbounded oscillating production of red cells.