  In this article, we study conditions of continuous emergent symmetries in
gapless states, either as topological quantum critical points (TQCPs) or a
stable phase with protecting symmetries and connections to smooth deformations
of the gapped states around. We illustrate that for a wide class of gapless
states that can be associated with fully-isolated scale invariant fixed points,
there shall always be emergent continuous symmetries that are directly related
to smooth deformations of gapped states with symmetries lower than the
protecting ones $G_p$. For a 3D TQCP in DIII classes with $G_p=Z^T_2$,
$U_{EM}=U(1)$ and $N_f=\frac{1}{2}$ fermions but without charge $U(1)$
symmetry, we explicitly construct a corresponding boundary representation based
on a $4D$ topological state with lattice symmetry $H=Z^T_2 \ltimes U(1)$ and
$N_f={1}$ fermions. Although emergent continuous symmetries appear to be robust
at weakly interacting TQCPs, we further show the breakdown of such one-to-one
correspondence between deformations of gapped states and emergent continuous
symmetries when gapless states become strongly interacting. In a strongly
interacting limit, gapless states can be represented by a smooth manifold of
conformal-field-theory fixed points rather than a fully isolated one. A smooth
manifold of strong coupling fixed points hinders emergence of a continuous
emergent symmetry in the strongly interacting gapless limit, as deformations no
longer leave a gapless state or a TQCP invariant, unlike in the more
conventional weakly interacting case. This typically reduces continuous
emergent symmetries to a discrete symmetry originating from duality
transformations under the protection symmetry $G_p$.
