 Solution

For any real smooth function $\chi (x)$, make the local transformationThe electromagnetic field tensor is unchanged because the mixed derivatives of $\chi$ commute. Directly,Its conjugate transforms with the inverse phase, so $(D_{\mu }\phi {)}^{\ast }{D}^{\mu }\phi$ is invariant. The mass term depends only on $|\phi {|}^2$ and is invariant as well. All terms of the action are therefore unchanged. This proves the local U(1) gauge symmetry of scalar electrodynamics.