  The improper ferroelectricity in YMnO$_3$ and other related multiferroic
hexagonal manganites are known to cause topologically protected ferroelectric
domains that give rise to rich and diverse physical phenomena. The local
structure and structural coherence across the ferroelectric transition,
however, were previously not well understood. Here we reveal the evolution of
the local structure with temperature in YMnO$_3$ using neutron total scattering
techniques, and interpret them with the help of first-principles calculations.
The results show that, at room temperature, the local and average structures
are consistent with the established ferroelectric $P6_3cm$ symmetry. On
heating, both local and average structural analyses show striking anomalies
from $\sim 800$ K up to the Curie temperature consistent with increasing
fluctuations of the order parameter angle. These fluctuations result in an
unusual local symmetry lowering into a \textit{continuum of structures} on
heating. This local symmetry breaking persists into the high-symmetry non-polar
phase, constituting an unconventional type of order-disorder transition.
