In this paper, we study the concept of soft sets which is introduced by Molodtsov  and the notion of soft continuity is introduced by Zorlutuna et al. . We give the definition of $(\tau_1,\tau_2)$ - semi open soft ( resp. $(\tau_1,\tau_2)$ - pre open soft, $(\tau_1,\tau_2)-\alpha$ - open soft, $(\tau_1,\tau_2)-\beta$ - open soft) set via two soft topologies. We introduce mixed semi - soft (resp. mixed pre - soft, mixed $\alpha$ - soft, mixed $\beta$ - soft) continuity between two soft topological spaces $(X,\tau_1,A),(X,\tau_2,A)$ and a soft topological space $(Y,\tau,B)$. Also we prove that a function is mixed $\alpha$ - soft continuous if and only if it is both mixed pre - soft continuous and mixed semi - soft continuous.