  It is proved that an inhomogeneous medium whose boundary contains a weakly
singular point of arbitrary order scatters every incoming wave. Similarly, a
compactly supported source term with weakly singular points on the boundary
always radiates acoustic waves. These results imply the absence of
non-scattering energies and non-radiating sources in a domain that is not
$C^\infty$-smooth. Local uniqueness results with a single far-field pattern are
obtained for inverse source and inverse medium scattering problems. Our
arguments provide a sufficient condition of the surface under which solutions
to the Helmholtz equation admits no analytical continuation.
