condensed matter - Can gapped state and gapless state be adiabatically connected to each other?

Is it possible that we can construct a gapped state and a gapless state which are adiabatically connected?

Here adiabatically connected I mean:

there exists a class of Hamiltonians $H(g)$ with ground state $|\phi(g)>$($g\in[0,1]$), such that $|\phi(0)>$ is gapless and $|\phi(1)>$ is gapped. And the ground state average of any local operator $(g)$ doesn't has singularity for all $g\in[0,1]$

1. 
   

   If it's possible, can some one give me an example?

   
   


2. 
   

   If it's impossible. Does it imply we can always find a topological order or a normal order parameter to distinguish a gapped phase from a gapless phase.