According to one of the properties established before for the first-order filter [5], if is continuous in the domain where the filter is applied, then is differentiable, and its derivative is given by
This is valid so long as the support interval of the filter fits completely inside the region where is continuous. This immediately implies that, in a region where increases monotonically we have
We may therefore conclude that also increases monotonically within the sub-region where the support interval of the filter fits inside the region in which is continuous. In the same way, in a region where decreases monotonically we have
We may therefore conclude that also decreases monotonically within that same sub-region. In other words, the monotonic character of the variation of a function is invariant by the action of the filter. In particular, at points where is differentiable the sign of its derivative is invariant by the action of the filter.