Recall that traditional Morse Theory begins with this insight [4]: Let f (usually called a height function) be a C2 continuous function defined on a compact, smoothly differentiable manifold M. A Morse function has the following properties: Each critical point of f is an isolated point, and at each critical point the Hessian (matrix of second order partials) is nonsingular. In other words each criticality is a single isolated point and is a true local maximum, minimum or saddle point (there are no points of inflection). Then the topology changes of the level sets of f occur only at the critical values and are completely characterized by the number of negative eigenvalues of the Hessian at each critical point, which determines the number of linearly independent down directions, and thus whether it is a maximum, a minimum, or determines the type of saddle.
We apply this insight to our discrete setting[5], where
we cannot assume that the function is Morse in the aforementioned sense.
This is because the data may contain clusters of identically valued readings,
and thus, in general, not all critical points can be assumed isolated.
Since our data readings (function 
)
can be extended to a continuous function f in
uncountably many ways, we have to make some base assumptions, and these assumptions
are consistent with the majority of the current practice and literature on so-called volumetric or density data.