Let M be the set of all functions meromorphic on D={z∈ℂ:|z|<1}.
For a∈(0,1], a function f∈M is called a-normal function of bounded (vanishing) type or
f∈Na(N0a), if supz∈D(1−|z|)af#(z)<∞ (lim|z|→1(1−|z|)af#(z)=0). In this paper
we not only show the discontinuity of Na and N0a relative to containment as a varies, which
shows ∪0<a<1Na⊂UBC0, but also give several characterizations of Na and N0a which are real
extensions for characterizations of N and N0.