= Solution
For any real smooth function $\chi(x)$, make the local transformation
$$
\boxed{\varphi'(x)=e^{-ie\chi(x)}\varphi(x),\qquad A'_\mu=A_\mu+\partial_\mu\chi.}
$$
The <electromagnetic field tensor> is unchanged because the mixed derivatives of $\chi$ commute. Directly,
$$
(\partial_\mu+ieA'_\mu)\varphi'
=e^{-ie\chi}(\partial_\mu+ieA_\mu)\varphi.
$$
Its conjugate transforms with the inverse phase, so $(D_\mu\varphi)^*D^\mu\varphi$ is invariant. The mass term depends only on $|\varphi|^2$ and is invariant as well. All terms of the action are therefore unchanged. This proves the local <U(1) gauge symmetry> of <scalar electrodynamics>.
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