  Conventional diffractive and dispersive devices introduce angular dispersion
(AD) into pulsed optical fields thus producing so-called 'tilted pulse fronts'.
Naturally, it is always assumed that the functional form of the
wavelength-dependent propagation angle associated with AD is differentiable
with respect to wavelength. Recent developments in the study of space-time wave
packets -- pulsed beams in which the spatial and temporal degrees of freedom
are inextricably intertwined -- have pointed to the existence of
non-differentiable AD: field configurations in which the propagation angle does
not possess a derivative at some wavelength. Here we investigate the
consequences of introducing non-differentiable AD into a pulsed field and show
that it is the crucial ingredient required to realize group velocities that
deviate from $c$ (the speed of light in vacuum) along the propagation axis in
free space. In contrast, the on-axis phase and group velocities are always
equal in conventional scenarios. Furthermore, we show that non-differentiable
AD is needed for realizing anomalous or normal group-velocity dispersion along
the propagation axis, while simultaneously suppressing all higher-order
dispersion terms. These and several other consequences of non-differentiable AD
are verified experimentally using a pulsed-beam shaper capable of introducing
AD with arbitrary spectral profile. Rather than being an exotic phenomenon,
non-differentiable AD is an accessible, robust, and versatile resource for
sculpting pulsed optical fields.
