Recently in relation to the theory of non-commutative probability, the notion of evolution family
{
ω
s
,
t
}
s
≤
t
\{\omega _{s,t}\}_{s \le t}
is generalized assuming only continuity in parameters, namely
(
s
,
t
)
↦
ω
s
,
t
(s,t) \mapsto \omega _{s,t}
is continuous with respect to locally uniform convergence on a planar domain. In this article we present various equivalence conditions to the continuous evolution families concerned with the left and right parameters. We also provide an example of a discontinuous evolution family in the last section.