When Witkin  introduced the term scale-space,
he was concerned with one-dimensional signals and observed
that new local extrema cannot be created in this family.
Since differentiation commutes with convolution,
this non-creation property applies also to any -order spatial
derivative computed from the scale-space representation.
Figure 2(b) illustrates this property,
by showing zero-crossings of the second derivative of the
smoothed signal at different scales.
Note that the trajectories of zero-crossings in scale-space
form paths across scales that are never closed from below.
This property does, however, not extend to dimensions higher than one.