  In this work, we study an inverse problem of recovering information about the
weight in distributed-order time-fractional diffusion from the observation at
one single point on the domain boundary. In the absence of an explicit
knowledge of the medium, we prove that the one-point observation can uniquely
determine the support bound of the weight. The proof is based on asymptotics of
the data, analytic continuation and Titchmarch convolution theorem. When the
medium is known, we give an alternative proof of an existing result, i.e., the
one-point boundary observation uniquely determines the weight. Several
numerical experiments are also presented to complement the analysis.
